Optimization of NMR Experiments using Genetic Algorithm: …spinhalf.in/bio_manu/manu_Thesis.pdf · 2019-02-04 · I thank Prof. Apoorva Patel for his introductory courses on Quantum

Optimization of NMR Experiments using Genetic

Algorithm: Applications in Quantum Information

Processing, Design of Composite Operators and

Quantitative Experiments

A Thesis

Submitted For the Degree of

Doctor of Philosophy

in the Faculty of Science

by

V. S. MANU

Department of Physics

Indian Institute of Science

BANGALORE – 560 012

December 2013

Declaration

I hereby declare that the work reported in this thesis is entirely original. It was

carried out by me in the Department of Physics, Indian Institute of Science, Ban-

galore. I further declare that it has not formed the basis for the award of any

degree, diploma, membership, associateship or similar title of any university or

institution.

December 15, 2013 (V. S. MANU)

iii

“I study myself more than any other subject. That is my metaphysics, that is my

physics.”

Michel de Montaigne

Publications based on this Thesis

1. Non-Destructive Discrimination of Arbitrary Set of Orthogonal Quantum

States by NMR using Quantum Phase Estimation,

V. S. Manu and Anil Kumar,

AIP Conference Proceedings. 1384 229(2011).

2. Singlet-state Creation and Universal Quantum Computation in NMR using

a Genetic Algorithm,


Phys. Rev. A 86 022324(2012).

3. Fast and Accurate Quantification using Genetic Algorithm Optimized 1H-13C Refocused Constant-Time INEPT,


Journal of Magnetic Resonance 234 106 (2013).

4. Quantum Simulation using Fidelity Profile Optimization,


Phys. Rev. A (Under Review).

5. Efficient Creation of NOON States in NMR,

V. S. Manu and Anil Kumar (To be communicated),

6. Composite Operator Design in NMR using Genetic Algorithm,

V. S. Manu and Anil Kumar (To be communicated).

vii

Abstract

Genetic algorithms (GA) are stochastic global search methods based on the me-

chanics of natural biological evolution, proposed by John Holland in 1975. Here

in this thesis, we have exploited possible utilities of Genetic Algorithm optimiza-

tion in Nuclear Magnetic Resonance (NMR) experiments. We have performed

(i) Pulse sequence generation and optimization for NMR Quantum Information

Processing, (ii) efficient creation of NOON states, (iii) Composite operator design

and (iv) delay optimization for refocused quantitative INEPT. We have generated

time optimal as well as robust pulse sequences for popular quantum gates. A

Matlab package is developed for basic Target unitary operator to pulse sequence

optimization and is explained with an example.

Chapter 1 contains a brief introduction to NMR, Quantum computation and Ge-

netic algorithm optimization. Experimental unitary operator decomposition using

Genetic Algorithm is explained in Chapter 2. Starting from a two spin homonu-

clear system (5-Bromofuroic acid), we have generated hard pulse sequences for

performing (i) single qubit rotation, (ii) controlled NOT gates and (iii) pseudo

pure state creation, which demonstrates universal quantum computation in such

systems. The total length of the pulse sequence for the single qubit rotation of an

angle π/2 is less than 500µs, whereas the conventional method (using a selective

soft pulse) would need a 2ms shaped pulse. This substantial shortening in time

can lead to a significant advantage in quantum circuits. We also demonstrate the

creation of Long Lived Singlet State and other Bell states, directly from thermal

equilibrium state, with the shortest known pulse sequence. All the pulse sequences

generated here are generic i.e., independent of the system and the spectrometer.

We further generalized this unitary operator decomposition technique for a vari-

able operators termed as Fidelity Profile Optimization (FPO) (Chapter 3) and

performed quantum simulations of Hamiltonian such as Heisenberg XY interaction

and Dzyaloshinskii-Moriya interaction. Exact phase (φ) dependent experimental

unitary decompositions of Controlled-φ and Controlled Controlled-φ are solved

using first order FPO. Unitary operator decomposition for experimental quantum

simulation of Dzyaloshinskii-Moriya interaction in the presence of Heisenberg XY

interaction is solved using second order FPO for any relative strengths of interac-

tions (γ) and evolution time (τ). Experimental gate time for this decomposition

ix

is invariant under γ or τ , which can be used for relaxation independent studies of

the system dynamics. Using these decompositions, we have experimentally veri-

fied the entanglement preservation mechanism suggested by Hou et al. [Annals of

Physics, 327 292 (2012)].

NOON state or Schrodinger cat state is a maximally entangled N qubit state

with superposition of all individual qubits being at |0〉 and being at |1〉. NOON

states have received much attention recently for their high precession phase mea-

surements, which enables the design of high sensitivity sensors in optical interfer-

ometry and NMR [Jones et al. Science, 324 1166(2009)]. We have used Genetic

algorithm optimization for efficient creation of NOON states in NMR (Chapter 4).

The decompositions are, (i) a minimal in terms of required experimental resources

– radio frequency pulses and delays – and have (ii) good experimental fidelity.

A composite pulse is a cluster of nearly connected rf pulses which emulate the

effect of a simple spin operator with robust response over common experimental

imperfections. Composite pulses are mainly used for improving broadband de-

coupling, population inversion, coherence transfer and in nuclear overhauser effect

experiments. Composite operator is a generalized idea where a basic operator

(such as rotation or evolution of zz coupling) is made robust against common

experimental errors (such as inhomogeneity / miscalibration of rf power or er-

ror in evaluation of zz coupling strength) by using a sequence of basic operators

available for the system. Using Genetic Algorithm optimization, we have designed

and experimentally verified following composite operators, (i) broadband rotation

pulses, (ii) rf inhomogeneity compensated rotation pulses and (iii) zz evolution

operator with robust response over a range of zz coupling strengths (Chapter 5).

We also performed rf inhomogeneity compensated Controlled NOT gate.

Extending Genetic Algorithm optimization in classical NMR applications, we have

improved the quantitative refocused constant-time INEPT experiment (Q-INEPT-

CT) of Makela et al. [JMR 204(2010) 124-130] with various optimization con-

straints. The improved ‘average polarization transfer’ and ‘min-max difference’

of new delay sets effectively reduces the experimental time by a factor of two

(compared with Q-INEPT-CT, Makela et al.) without compromising on accuracy

(Chapter 6). We also introduced a quantitative spectral editing technique based

on average polarization transfer. These optimized quantitative experiments are

also described in Chapter 6.

Time optimal pulse sequences for popular quantum gates such as, (i) Controlled

Hadamard (C-H) gate, (ii) Controlled-Controlled-NOT (CCNOT) Gate and (iii)

Controlled SWAP (C-S) gate are optimized using Genetic Algorithm (Appendix.

A). We also generated optimal sequences for Quantum Counter circuits, Quan-

tum Probability Splitter circuits and efficient creation of three spin W state. We

have developed a Matlab package based on GA optimization for three spin target

operator to pulse sequence generator. The package is named as UOD (Unitary

Operator Decomposition) is explained with an example of Controlled SWAP gate

in Appendix. B.

An algorithm based on quantum phase estimation, which discriminates quantum

states non-destructively within a set of arbitrary orthogonal states, is described

and experimentally verified by a NMR quantum information processor (Appendix.

C). The procedure is scalable and can be applied to any set of orthogonal states.

Scalability is demonstrated through Matlab simulation.

Acknowledgements

Completion of this doctoral dissertation was possible with the support of several

people. I would like to express my sincere gratitude to all of them. Foremost, I

would like to express my sincere gratitude to my advisor Prof. Anil Kumar, who

has supported me throughout my thesis with his patience, immense knowledge

and for allowing me to work in my own way. His guidance helped me in all the

time of research and writing of this thesis. He has always made himself available

to clarify my doubts despite his busy schedules. I would like to thank Prof. Rahul

Pandit for his inspirational advises and motivation as my research supervisor. One

could not wish for a friendlier supervisor and I am blessed with two.

I thank Prof. K. V. Ramanathan for his beautiful way of introducing the world

of Nuclear Magnetic Resonance and for numerous fruitful discussions. I would

like to thank Prof. N. Suryaprakash for introductory lectures on NMR. He is ever

smiling and always ready to help. I humbly acknowledge Dr. S. Raghothama and

Dr. Athreya for their basic NMR courses, which helped me a lot in understanding

the NMR methodologies. I thank Mr. P. T. Wilson for helping in solving problems

with NMR spectrometers.

I thank Prof. Apoorva Patel for his introductory courses on Quantum Computa-

tion and clarifying the concepts in Quantum world. I thank Prof. P. C. Mathias

and Vincent for the efficient maintenance of the spectrometers which helped a lot

in saving time. I thank Chandru for his help in spectrometer problems.

I would like to thank T. S. Mahesh for constructive criticisms and valuable sug-

gestions. I would like to acknowledge Koteswar, my only lab mate for his support

and encouragements. I like to thank Krishnarjuna for many focused discussions

and encouragements.

In my daily work I have been blessed with a friendly and cheerful group of lab

mates. I gratefully acknowledge my seniors Avik Mithra, Bibhuthi, Jharana,

Jayanthi, Nithin Lobo and Uday and Sankeerth for their help in understand-

ing NMR. I thank Nilamoni, Monalisa, Garima, Shibdas Roy, Dr. Bolanath, U.

V. Reddy, Subbareddy, Rajesh Sonti, Thirupathi, Kowsalya, Sudheer, Shivanand,

Sachin, Lokesh, Somnath, Pratik, Srinu, Vasu,Venkat, Santhosh and Durgesh for

numerous academic and non-academic discussions I had with them and for making

my stay here memorable.

xiii

I would like to thank Raghav G. Mavinkurve, an all-rounder helping hand, through-

out my Ph.D life. His way of handling problem is an inspiration to me all the time.

I would like to thank all my friends Alex, Akhil, Arunbabu, Ben, Naveen (Yeshu),

Pradeep, Rajesh, Ranjith, Sajesh, Sanjay, Simil, Vineeth, Vivek, Aneesh, Anoop,

Eby and Joji for many unforgettable moments I shared with them.

Past and present project assistants of NMR Research Center are also gratefully

acknowledged for their smooth running of the spectrometers. Thanks to all Physics

and NRC office staffs, who have helped me effortlessly get through the official

formalities.

I would like to thank Biju sir and Mathew sir (BSc teachers) for their constant

support and encouragement.

I wish to thank my parents, sister, brother in-law and nephews (Kannan & Kunju)

for their moral support and love. Their love was my driving force during my Ph.D

life. I owe them everything and wish I could show them just how much I love and

appreciate them.

Finally but not the least I would like to acknowledge Department of Physics and

NMR Research Centre for providing the support and equipment I have needed to

produce and complete my thesis. UGC and CQIQC are acknowledged for financial

support.

Contents

Declaration of Authorship iii

Publications based on this Thesis vii

Abstract ix

Acknowledgements xiii

List of Figures xix

List of Tables xxiii

Abbreviations xxv

Physical Constants xxvii

1 Introduction 1

1.1 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Interactions in NMR . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Thermal Equilibrium state in NMR (ρeq) . . . . . . . . . . . 6

1.2 Quantum Information Processing (QIP) . . . . . . . . . . . . . . . 7

1.2.1 NMR Quantum Information Processing . . . . . . . . . . . . 10

1.2.1.1 Pseudo Pure State Creation . . . . . . . . . . . . . 11

1.2.1.2 Quantum State Tomography (QST) . . . . . . . . 12

1.3 Genetic Algorithm optimization . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Population Initialization . . . . . . . . . . . . . . . . . . . . 17

1.3.2 Selection Operators . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.3 Reproduction Operators . . . . . . . . . . . . . . . . . . . . 18

1.3.3.1 Crossover Operators: . . . . . . . . . . . . . . . . . 18

1.3.3.2 Mutation Operators . . . . . . . . . . . . . . . . . 19

2 Genetic Algorithm Optimization in NMR Quantum InformationProcessing 23

xv

Contents xvi

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Genetic Algorithm for NMR Pulse Sequence Generation . . . . . . . 24

2.2.1 Representation Scheme . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Fitness Function . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Two qubit homonuclear case . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Operator Optimization . . . . . . . . . . . . . . . . . . . . . 26

2.3.1.1 Single Qubit Rotations using non-selective pulses . 27

2.3.1.2 Controlled NOT gate . . . . . . . . . . . . . . . . . 28

2.3.2 State to State Optimization . . . . . . . . . . . . . . . . . . 30

2.3.2.1 Pseudo-Pure State Creation . . . . . . . . . . . . . 30

2.3.2.2 Creation of Bell States Directly from the ThermalState . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Quantum Simulation Using Fidelity Profile Optimization 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Theory - Fidelity Profile Optimization (FPO) . . . . . . . . . . . . 39

3.2.1 First Order FPOs . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1.1 Controlled-Phase gate (C-φ) . . . . . . . . . . . . 41

3.2.1.2 Controlled-Controlled-Phase gate (CC-φ) . . . . . 41

3.2.1.3 Quantum Simulation of three spin Heisenberg XYHamiltonian . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1.4 Three Spin DM interaction . . . . . . . . . . . . . 43

3.2.1.5 Adiabatic Quantum Evolution . . . . . . . . . . . . 44

3.2.2 Second order FPO . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.1 Controlled Phase (C-φ) gate . . . . . . . . . . . . . . . . . . 51

3.3.2 Quantum Simulation of three spin Heisenberg XY Hamiltonian 51

3.3.3 Quantum simulation of DM Hamiltonian and entanglementpreservation . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Efficient Creation of NOON States in NMR 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 NOON State Creation (NSC) operators . . . . . . . . . . . 58

4.2.1.1 Spin chain with nearest neighbor couplings . . . . . 60

4.2.1.2 Spin star topology . . . . . . . . . . . . . . . . . . 61

4.2.2 Disentangling NOON states . . . . . . . . . . . . . . . . . . 62

4.2.3 Robust NOON state creation . . . . . . . . . . . . . . . . . 62

4.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Composite Operator Design Using Genetic Algorithm 69

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Theory and Experimental . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.1 Representation Scheme . . . . . . . . . . . . . . . . . . . . . 72

Contents xvii

5.2.2 Fitness Function . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.3 Broadband excitation pulses . . . . . . . . . . . . . . . . . . 74

5.2.4 Broadband Inversion pulses . . . . . . . . . . . . . . . . . . 75

5.2.5 Broadband π/2 pulses . . . . . . . . . . . . . . . . . . . . . 77

5.2.6 Broadband π pulses . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.7 rf inhomogeneity compensated composite pulses . . . . . . . 81

5.2.8 Composite Polarization transfer and Evolution operator . . . 83

5.2.9 RF inhomogeneity compensated Controlled NOT gate . . . . 86

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6 Fast and Accurate Quantification Using GA Optimized 1H −13 CRefocused Constant-Time INEPT 89

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2.1 Near uniform and equal response for CH1, CH2 and CH3

transfer functions . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.2 Near uniform and separate response for CH1, CH2 and CH3

transfer functions . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2.3 Near uniform and separate response for CH2 and CH3 only 94

6.2.4 Spectral editing by delay sets . . . . . . . . . . . . . . . . . 96

6.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

A Unitary Decomposition of Quantum Gates Using GA Optimiza-tion 103

A.1 Controlled Hadamard (C-H) . . . . . . . . . . . . . . . . . . . . . . 103

A.2 Controlled-Controlled-NOT (CCNOT) Gate . . . . . . . . . . . . . 104

A.3 Controlled SWAP (C-S) . . . . . . . . . . . . . . . . . . . . . . . . 105

A.4 Quantum Counter circuits . . . . . . . . . . . . . . . . . . . . . . . 106

A.5 W State Creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A.6 Quantum Probability Splitter (QPS) . . . . . . . . . . . . . . . . . 110

B Matlab UOD Package 113

C Non-Destructive Discrimination of Arbitrary Set of OrthogonalQuantum States by NMR Using Quantum Phase Estimation 139

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

C.2.1 The General Procedure (n-qubit case): . . . . . . . . . . . . 140

C.2.2 Single qubit case: . . . . . . . . . . . . . . . . . . . . . . . . 142

C.2.3 Two qubit case: . . . . . . . . . . . . . . . . . . . . . . . . . 143

C.2.3.1 Special case (α = β = 1√2): . . . . . . . . . . . . . . 145

C.2.4 Bell state discrimination: . . . . . . . . . . . . . . . . . . . . 145

C.2.5 GHZ state discrimination: . . . . . . . . . . . . . . . . . . . 146

C.3 Experimental Implementation by NMR . . . . . . . . . . . . . . . . 147

Contents xviii

C.3.1 Non-Destructive Discrimination of two qubit orthogonal states:147

C.3.1.1 Implementation of Controlled-U1 and U2: . . . . . 149

C.4 Three qubit GHZ state Discrimination using Matlab® Simulation: 151

C.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Bibliography 157

Index 171

List of Figures

1.1 Quantization of Spin 1/2 nucleus in presence of an applied field B0. 2

1.2 Energy Level diagram of a spin 1/2 nucleus in applied field B0. . . . 3

1.3 Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Quantum circuit representation of Controlled-NOT gate (Eqn. 1.29). 9

1.5 Function profile of g(x1, x2) (Eqn. 1.35). Maximas (mx1, mx2 andmx3) and minimas (mn1) of the function are marked. . . . . . . . . 14

1.6 Mechanism of biological evolution. . . . . . . . . . . . . . . . . . . . 15

1.7 Basic outline of Genetic Algorithm. . . . . . . . . . . . . . . . . . . 17

1.8 Roulette Wheel selection method for a popuation of six individualsA, B, C, D, E, F. The area for each individual is proportional toits fitness value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.9 Three different types of Children from existing generation to newgeneration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.10 Different Crossover operators. . . . . . . . . . . . . . . . . . . . . . 20

2.1 A two-spin system with chemical shifts ±δ and coupling J . . . . . 26

2.2 Pulse sequence for single qubit rotation. . . . . . . . . . . . . . . . 27

2.3 (a) Chemical structure of 5-Bromofuroic acid, (b) Equilibrium spec-trum, (c) (π/2)y SQR pulse on spin-1. (d) (π/2)y SQR pulse onspin-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Matlab simulation study of fidelity variation of SQR with differentvalues of J/δ and θ. The special case for 5-Bromofuroic acid (J/δ =0.014) is shown with an arrow in x axis. . . . . . . . . . . . . . . . 30

2.5 (a) Pulse sequence for CNOT gate, (b) Diagonal element tomog-raphy of: (i) equilibrium state, and states obtained after applying(ii) C1NOT2, (iii) C2NOT1, (iv) C1NOT2, (v) C2NOT1, (c) Fi-delity (F ) vs. (J/δ) plot for CNOT gate. . . . . . . . . . . . . . . . 31

2.6 (a) Pulse sequence for PPS creation, (b) Tomography of the diag-onal elements after preparing (i) |00〉 PPS, (ii) |01〉 PPS, (iii) |10〉,and (iv) |11〉 PPS. (c) Theoretical fidelity (F ) vs. (J/δ) plot of PPSgeneration pulse sequence. . . . . . . . . . . . . . . . . . . . . . . . 33

2.7 (a) Pulse sequence for creating the Bell states directly form thethermal equilibrium state, (b) Observation of the singlet state afterapplying the operator U of Eq. 2.9, (c) Density matrix tomographyof the created singlet state. . . . . . . . . . . . . . . . . . . . . . . . 34

2.8 Anti-phase signal decay as a function of interval and fits to a singleexponential decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xix

List of Figures xx

3.1 Fidelity profile of FPO (Eqn. 3.12) of Heisenberg XY interaction(Eqn. 3.11). Fidelities are calculated using Eqn. 3.3. . . . . . . . . 43

3.2 Pulse sequence for experimental simulation of three spin DM inter-action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Pulse Sequence decomposition of Eqn. 3.18. . . . . . . . . . . . . . 45

3.4 Fidelity profile of the Hamiltonian (Eqn. 3.18). . . . . . . . . . . . 45

3.5 Fidelity profile of UOD given in Eqn. 3.24. (Fidelities are calculatedusing Eqn. 3.3.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6 13C labeled Chloroform, two qubit NMR system, used for the quan-tum simulation of DM interaction (Eqn. 3.32) and controlled phasegate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.7 13C labeled CHFBr2 used for experimental verification of FPO ofHeisenberg XY interaction. . . . . . . . . . . . . . . . . . . . . . . . 49

3.8 Pulse sequece for two spin pesudo pure state creation. . . . . . . . . 51

3.9 Experimental implementation of Controlled Phase (φ) gate for var-ious phase angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.10 Experimental quantum simulation of Heisenberg XY Hamiltonian. . 52

3.11 Experimental scheme used for entanglement preservation. . . . . . . 54

3.12 (a).Entanglement dynamics of 13C −H system under the Hamilto-nian Eqn. 3.21, (b). Entanglement preservation experiment usingEqn. 3.37. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 Quantum circuit for NOON state creation. N qubit system requires(N − 1) Controlled-NOT gates and a Hadamard gate. . . . . . . . . 59

4.2 Pulse sequence for for two spin NOON state creation. . . . . . . . . 61

4.3 (a). Spin ising chain with nearest neighbor couplings. J i,i+1 is thecoupling between ith and (i+ 1)th spin, (b). Spin Star network. AllB spins are coupled with spin A with equal strength and there is nocoupling between B spins, (c). A general multi-spin system (spincloud). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 (a). 90 composite pulse (five pulse) response for flip angle error,(b). 180 composite pulse (five pulse) response for flip angle error. . 64

4.5 Flip angle and coupling error response of composite evolution (Eqn.4.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.6 13C labeled CHFBr2 used for experimental NSC. The spins 1H, 13Cand 19F forms a linear spin chain with NN coupling values JH,Czz =224.5 Hz and JC,Fzz = −310.5 Hz, if the two bond JH,Fzz = 49.7 Hzis refocused during the entire experimental time. The longitudinaland transverse relaxation time – T1, T2 (in seconds)– for 1H, 13Cand 19F are 6.7, 1.4, 1.9, 0.71 and 4.0, 0.7. . . . . . . . . . . . 65

4.7 Pulse sequece for three spin pesudo pure state creation. . . . . . . . 66

List of Figures xxi

4.8 Quantum state tomography of three spin NOON state created in(a). spin chain (Eqn. 4.11) and (b). spin star topology (Eqn.4.13) in combination with composite evolution sequence shown inEqn. 4.20. The labels in XY plane 1,2...8 correponds to thecomputational basis states |000〉, |001〉, ...... |111〉. The observedexperimental fidelity is 95.9 % for spin chain and 96.4 % for spinstar topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.1 Computational time advantage (γ, Eqn. 5.7) efficiency of effec-tive operator evaluation using ‘CompositePulseOperator ’ and Mat-lab builtin function ‘expm’ [1]. The response is evaluated for a setof randomnly created N pulse composite sequence. (Matlab 7.14,32 bit Ubuntu 12.04 OS with core i5 3.6 GHz Intel processor) . . . 71

5.2 Matlab simulation of three pulse, five pulse and seven pulse broad-band excitation sequences. (State to state optimization with Iz asinitial state and Ix as final state.) . . . . . . . . . . . . . . . . . . . 75

5.3 Experimental proton spectral intensity variation with different off-set values of three pulse, five pulse and seven pulse broadband ex-citation sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 Matlab simulation of three pulse, five pulse and seven pulse broad-band inversion sequences. (State to state optimization with Iz asinitial state and −Iz as final state.) . . . . . . . . . . . . . . . . . . 77

5.5 Experimental proton spectral intensity variation with different off-set values of three pulse, five pulse and seven pulse broadband in-version sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.6 Matlab simulation of three pulse, five pulse and seven pulse broad-band π/2 sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.7 Experimental proton spectral intensity variation with different off-set values of three pulse, five pulse and seven pulse broadband π/2sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.8 Matlab simulation of three pulse, five pulse and seven pulse broad-band π sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.9 Experimental proton spectral intensity variation with offset valuesof broadband three pulse, five pulse and seven pulse π compositepulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.10 Matlab simulation of seven pulse and nine pulse composite excita-tion sequences with robust rf inhomogeneity profile. . . . . . . . . . 83

5.11 Matlab simulation of five pulse π/2 and π composite sequences withrobust rf inhomogeneity profile. . . . . . . . . . . . . . . . . . . . . 84

5.12 Experimental proton spectral intensity variation with rf values offive pulse π/2 and seven pulse excitation. . . . . . . . . . . . . . . . 84

5.13 Composite operator sequence for for J compensation. . . . . . . . . 85

5.14 Pulse Sequence for Controlled NOT gate. . . . . . . . . . . . . . . . 86

5.15 Inhomogeneity response profiles of controlled NOT gate. . . . . . . 86

6.1 GAQIC pulse sequence. . . . . . . . . . . . . . . . . . . . . . . . . . 92

List of Figures xxii

6.2 Comparison of polarization transfer response of refocused INEPTand GAQIC experiments. . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3 Polarization transfer responses for quantitative spectral editing de-lay sets. (a). four scan (S4 ), (b). sixteen scan (S16 ) . . . . . . . . 97

6.4 Angle sets (θ = Jτ) for (a) SI1, (b) SI2 and (c) SI3. . . . . . . . . 98

6.5 Samples used for GAQIC experiment (a) 4-Penten 2 ol (P), (b) α-methoxy phenyl acetic acid (M) and (c) 1-2-dimethyl cyclohexane(D). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.6 Equilibrium spectrum of the compound mixture (Fig. 6.5) . . . . . 99

6.7 Experimental implementation of GAQIC spectral editing delay set.(a). 1H decoupled 13C spectra and (b). using the delay set (S4). . . 100

A.1 Pulse sequence for Controlled Hadamard. . . . . . . . . . . . . . . . 104

A.2 Pulse sequence for Toffoli Gate. . . . . . . . . . . . . . . . . . . . . 105

A.3 Controlled-SWAP decomposition using C-NOT gates. . . . . . . . . 106

A.4 Pulse sequence for Controlled SWAP gate. . . . . . . . . . . . . . . 107

A.5 Three qubit quantum counter using CC-NOT and C-NOT Gates. . 107

A.6 Pulse sequence for Two Qubit Counter. . . . . . . . . . . . . . . . . 108

A.7 Pulse sequence for Three Qubit Counter. . . . . . . . . . . . . . . . 109

A.8 Pulse sequence for W state creation. . . . . . . . . . . . . . . . . . 110

A.9 Quantum circuit for probability splitter. . . . . . . . . . . . . . . . 110

A.10 Pulse sequence for Quantum Probability Splitter. . . . . . . . . . . 110

B.1 Controlled SWAP pulse sequence. . . . . . . . . . . . . . . . . . . 138

C.1 The general circuit for non-destructive Quantum State Discrimina-tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.2 (a) Two qubit State discrimination circuit for Experimental im-plementation in three qubit NMR quantum computer, (b) and (c)are splitting of the circuit-(a) into two circuits with single ancillameasurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

C.3 The three qubit NMR sample used for experimental implementation.149

C.4 The pulse sequence for Controlled-U1 and Controlled-U2 operatorsfor two qubit orthogonal states (Eqn. C.9) . . . . . . . . . . . . . . 150

C.5 Ancilla (proton spin) spectra of final state for two qubit state dis-crimination algorithm for (i) |φ1〉, (ii) |φ2〉, (iii) |φ3〉, (iv) |φ4〉states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

C.6 Density Matrix Tomography of the initial and final states of QSDcircuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

C.7 Pulse sequence for Controlled operators in GHZ state discrimination.153

C.8 The chemical structure, the chemical shifts and spin-spin couplingof a 13C labelled Crotonic Acid. The four 13C spins act as fourqubits [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

C.9 Matlab® simulation results for GHZ state discrimination. . . . . . 155

List of Tables

1.1 Spin quantum numbers of some of the common nuclei. . . . . . . . 2

1.2 Different physical implementation of quantum computer and crudeestimates of coherence time and gate operation time in seconds(from [3]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Darwinian Evolution and GA optimization problem comparison. . . 15

2.1 Spin selection and phase φ of the SQR is controlled by the phasesφ1 and φ2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 (θ,φ) values for all four CNOT gates. . . . . . . . . . . . . . . . . . 29

2.3 Values of φ and d for Bell state preparation using the pulse sequenceshown in Fig. 2.7(a). . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.1 Min-Max Difference (MMD) and Average Polarization Transfer (APT)of all optimized delay sets and R-INEPT. . . . . . . . . . . . . . . 96

6.2 Experimentally measured relative quantities of compound 2 (RQ21)and compound 3 (RQ22). . . . . . . . . . . . . . . . . . . . . . . . . 100

A.1 Phase modulation values for different Toffoli gates . . . . . . . . . . 105

A.2 Truth table for two qubit quantum counter. . . . . . . . . . . . . . 107

C.1 State of ancilla qubits for different input states for two qubit or-thogonal states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

C.2 State of ancilla qubits for different input states of Eqn. C.13 andC.19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

xxiii

Abbreviations

AED Average Experimental Deviation

APT Attached Proton Test / Average Polarization Transfer

CNOT Controlled NOT

DEPT Distortionless Enhancement by Polarization Transfer

DJ Deutsch - Jozsa

DM Dzyaloshinsky-Moriya

FPO Fidelity Profile Optimization

GA Genetic Algorithm

GAQIC Genetic Algorithm optimised Quantitative INEPT - Constant time

GRAPE Gradient Ascent Pulse Engineering

INEPT Insensitive Nuclei Enhanced by Polarization Transfer

LLSS Long Lived Singlet State

LOCC Local Operations and Classical Communication

LS-NMR Liquid State - Nuclear Magnetic Resonance

MMD Min-Max Difference

MRI Magnetic Resonance Imaging

NMR Nuclear Magnetic Resonance

NN Nearest Neighbor

NOE Nuclear Overhauser Effect

NSC NOON State Creation

PEA Phase Estimation Algorithm

PPS Pseudo Pure States

PFG Pulsed Field Gradients

QIP Quantum Information Processing

xxv

Abbreviations xxvi

QSD Quantum State Discrimination

QST Quantum State Tomography

SA Spatial Averaging

SQR Single Qubit Rotation

SMP Strongly Modulated Pulses

SS State to State

SSQB Single Superposed Qubits

SST Spin Star Topology

TA Temporal Averaging

UOD Unitary Operator Decomposition

Physical Constants

Planck constant h = 6.62606957× 1034 Js

reduced Planck constant = 1.054571726× 1034 Js

vacuum permeability µ0 = 4π × 10−7NA−2

Bohr magneton µB = 9.27400968× 10−24JT−1

Avogadro’s number NA, L = 6.02214129× 1023mol−1

Boltzmann constant kB = 1.3806488× 10−23JK−1

elementary charge e = 1.602176565× 10−19C

electron mass me = 9.10938291× 10−31kg

proton mass mp = 1.672621777× 10−27kg

xxvii

To my Parents,

xxix

Chapter 1

Introduction

The heroic age of quantum physics starts from Stern-Gerlach experiment in 1922.

Using a beam of silver atoms, Stern and Gerlach showed that measurable values of

atomic magnetic moments are discrete. Later in 1924, Wolfgang Pauli postulated

the existence of a nuclear spin to explain the hyperfine structure of atomic spectra.

The existence of a nuclear spin was confirmed experimentally by Rabi in 1939.

Rabi’s experiment showed that a beam of hydrogen molecules passing through a

magnetic field absorbs radio frequency of a discrete wavelength. 1943 and 1944

Physics Nobel prizes were awarded to Otto Stern and Isidor Isaac Rabi for these

findings[4].

The first NMR experiment on condensed matter was carried out by Felix Bloch

[5] and Edward Purcell[6], who were jointly awarded 1952 Nobel Prize in Physics

for their discovery. Fourier transform NMR was introduced in 1964 by Prof. R.R.

Ernst (chemistry Nobel Prize 1991 [4]). As a scientific tool, NMR is used for

determining the physical and chemical properties of molecules. This leads to a

wide range of applications in chemistry[7], biomolecular structural and functional

studies [8–12] and in medicine (MRI[13, 14], Functional MRI [15] and Magnetic

Resonance Microscopy [16]). Nuclear magnetic resonance is extremely useful since

it is a non destructive and noninvasive technique, which is very important in study-

ing biological systems and compounds [10]. Acquisition of dynamic information is

one of the remarkable advantages of NMR [17–20]. NMR spectroscopy is also an

important technique for studying time dependent chemical phenomena [21, 22].

1

Chapter 1. Introduction 2

Figure 1.1: Quantization of Spin 1/2 nucleus in presence of an applied fieldB0.

Element 1H 2H 12C 13C 14N 15N 16O 17O 19F 31P 35ClSpin 1

21 0 1

21 1

20 5

212

12

32

Table 1.1: Spin quantum numbers of some of the common nuclei.

1.1 Nuclear Magnetic Resonance

Many atomic nuclei possess an intrinsic form of angular momentum called ‘Spin’.

These spinning charge generates a magnetic field and the resulting spin-magnet

has a magnetic moment (µ) proportional to the spin I. Magnetic moment of these

atomic nuclei are quantized as per the quantization nature of angular momentum

operator, which is 2I + 1 discrete levels for a nuclear spin with spin quantum

number I. In the presence of an applied magnetic field, nuclear magnetic moment

associated with these discrete nuclear spin states experiences different magnetic

potential energies. This difference in energy between the spin states increases

linearly with the strength of the field. Irradiation of the sample with energy

corresponding to the exact spin state separation of a specific set of nuclei causes

excitation of those set of nuclei from the lower energy state to the higher energy

state. For a spin 1/2 nucleus the quantized states are spin up – aligns with the

magnetic field – and one spin down – opposes applied field (Fig. 1.1). Spin

quantum numbers of some common nuclei are given in Tab. 1.1.

1.1.1 Interactions in NMR

Atomic nuclei with non zero angular momentum (~I) behave like tiny magnets,

which interacts with applied or local magnetic fields. This interaction was first

observed by Pieter Zeeman as splitting a spectral line into several components in

the presence of a static magnetic field (1902 Nobel Prize). Magnetic moment of a


Figure 1.2: Energy Level diagram of a spin 1/2 nucleus in applied field B0.

nucleus with spin ~I is given by,

µ = γ~I, (1.1)

where γ is the gyromagnetic ratio and = h/2π, h is the Plank’s constant.

For a spin I = 1/2 nuclei, there are 2I + 1 = 2 discrete levels, which corresponds

to parallel and anti-parallel configuration to external magnetic field ~B(Fig. 1.1).

Both configuration experience different magnetic potential energies proportional

to strength of external field (B).

The Zeeman interaction Hamiltonian is given by,

H = −µ. ~B, (1.2)

where ~B is effective magnetic field vector experienced by the spin. For a constant

magnetic field along z direction ( ~B = B0z), Eqn. 1.2 can be written as,

Hz = −γB0Iz = ωIz, (1.3)

where Iz is the z component of ~I and ω is known as Larmor frequency (named

after Sir Joseph Larmor) of the nucleus. The energy level diagram of a spin 1/2

nucleus is shown in Fig. 1.2.

Atomic nuclei are shielded by the electrons that surrounded them. In the presence

of an applied magnetic field (B0), the induced magnetic field opposes the applied

field B0 and is known as diamagnetic shielding. This modifies the Hamiltonian

(Eqn. 1.3) as,

Hz = −γ(1− σ)B0Iz, (1.4)


where σ is the isotropic value of shielding tensor. Fast tumbling motion of molecules

in liquid state results in a time averaged Hamiltonian as,

σ =1

3(σxx + σyy + σzz). (1.5)

Depending on the surrounding electron density, the strength of diamagnetic shield-

ing will vary and cause a shift in the Larmor frequency of the nuclei, called Chem-

ical Shift (Ω).

The basic NMR interactions are,

(i) Interaction of spin with external field – Zeeman interaction Hz (Eqn. 1.4),

(ii) interaction between spins – scalar coupling (HJ) and dipolar coupling (HD),

(iii) interaction with electric-field gradient generated by their surroundings – for

spins with ~I > 1/2 – Quadrupolar coupling (HQ) and

(iv) interaction with time varying external rf field (Hrf ).

Hence most general NMR Hamiltonian is given by,

H = Hz +HJ +HD +HQ +Hrf . (1.6)

Scalar coupling or spin-spin coupling Hamiltonian (HJ) is an indirect interaction

mediated through electrons in the chemical bond between nuclei.

HJ = 2π∑l<m

~Il.Jlm.~Im, (1.7)

where Jlm is the spin-spin coupling tensor between lth and mth spins. In isotropic

case (liquid state NMR) only the trace of Jlm survives. Hence the spin-spin inter-

action is given by,

HJ = 2π∑l<m

JlmIl.Im, (1.8)

Under weak coupling approximation, where spin-spin coupling strength (Jlm) <<

chemical shift difference of spin l and m(ω0l − ω0m),

HwJ = 2π

∑l<m

JlmIzlIzm. (1.9)


Through space direct interaction of magnetic dipoles give rise the dipolar coupling

interaction, which has the form,

HD = 2π∑l<m

~Il.Dlm.~Im, (1.10)

where Dlm is the dipole tensor. For ~rlm, vector connecting two dipolar coupled

nuclei, l and m, Eqn. 1.10 can be expanded as,

HD =∑l<m

dlm[~Il.~Im − 3(~Il.rlm)(~Im.rlm)

r2lm

], (1.11)

where dlm = µ0γlγm/(4πr3lm).

For spins with angular momentum I > 1/2, interaction with electric field gradient

results in Quadrupolar interaction of the form,

HQ =∑l

~Il.Ql.~Il, (1.12)

where Ql is the quadrupolar coupling tensor. In terms of electric field gradient

tensor (Vl), at the site of spin l, Ql is given by,

Ql =eQl

2Il(2Il − 1)Vl, (1.13)

where Ql is the quadrupole moment of spin l.

In isotropic solution the dipolar and quadrupolar interactions average to zero in

the first order and do not contribute to the system Hamiltonian (Eqn. 1.6). The

interactions can be brought back by orienting the molecules in liquid crystal sol-

vents yielding reduced dipolar and quadrupolar couplings. However in this thesis,

we are dealing only with isotropic solutions.

The dynamics of the spin can be controlled externally using radio frequency pulses.

The spin Hamiltonian of isotropic solution during rf pulse is given by,

H = Hz +HJ +Hrf (t), (1.14)

where Hz = ω0Iz is the Zeeman Hamiltonian (ω0 = −γB0). And

Hrf = ω1Cos(ωrf t+ φ)Ix + Sin(ωrf t+ φ)Iy, (1.15)


where ω1 = −γB1 (B1 −→ amplitude of rf), ωrf is the frequency of applied rf

pulse with phase φ.

By a rotating frame transformation, using the operator [14],

U = eiωrf Izt, (1.16)

the rf Hamiltonian (Eqn. 1.15) can simplified to a time independent form, yielding

an effective Hamiltonian (Heff ) in the rotating frame as,

Heff = ΩIz + ω1(IxCosφ+ IySinφ), (1.17)

where Ω is the offset,

Ω = ω0(1− σ)− ωrf . (1.18)

1.1.2 Thermal Equilibrium state in NMR (ρeq)

A quantum spin system left undisturbed for a long time in contact with its sur-

rounding (or lattice) will reach a thermal equilibrium state (ρeq),

ρeq =e−H/kBT

Tr[e−H/kBT ], (1.19)

where kB (= 1.3806488 × 10−23JK−1) is the Boltzmann constant and T is the

lattice temperature. The coherence between any two quantum states in thermal

equilibrium is zero,

ρeqkl = 0, for k 6= l. (1.20)

The distribution of population of molecules in different energy levels of thermal

equilibrium state (diagonal elements of ρeq) follows Boltzmann statistics. At room

temperature and usual B0 fields, kBT becomes much larger than difference between

spin energy eigenvalues of H. In this high temperature approximation (upto first

order) of Eqn. 1.19 can be written as,

ρeq u1− H/kBT

Tr[1− H/kBT ], (1.21)

For an ensemble of N spins, the Eqn. 1.21 can be written as,

ρeq u1

2N(1− H

kBT). (1.22)


Substituting the Hamiltonian(H), Eqn. 1.22 can be simplified as,

ρeq u1

2N(1 +

B0

kBT

N∑i=1

γiIiz),

u1

2N(1 + ερdev),

(1.23)

where γi is the gyromagnetic ratio, B0 is the static external magnetic field and

ρdev is the deviation density matrix.

1.2 Quantum Information Processing (QIP)

Processing information stored in quantum states of a system, using quantum me-

chanical resources such as superposition and entanglement, can solve certain prob-

lems much faster than their classical counterparts. This made QIP, an intensively

investigated field in Physics over last few decades. The 2012 Nobel Prize in Physics

was awarded to Serge Haroche and David. J. Wineland “for ground breaking ex-

perimental methods that enable measuring and manipulation of individual quan-

tum systems”.

Quantum information unit in QIP is quantum bit (or qubit). Compared with

classical bit, which can assume only two states (either ‘0’ or ’1’), qubit can have

any superpositions of ‘0’ and ‘1’, yielding infinite number of quantum states. A

general single qubit state can be represented as,

|ψ〉 = Cos(θ/2)|0〉+ e−iφSin(θ/2)|1〉. (1.24)

The state (Eqn. 1.24) can be represented as a point in a unit sphere with position

(θ, φ), known as Bloch sphere(Fig. 1.3). All the points in the surface of Bloch

sphere are pure states and those inside are mixed state. A general single qubit

density matrix (ρ) can be represented as,

ρ =1

2(1 + ~r.~σ), (1.25)

where ~r is Bloch vector and ~σ = σx, σy, σz. The magnitude of Bloch vector

(r = |~r|) is unity for pure states and remains unchanged during unitary evolutions.

For a mixed state Trace(ρ2) < 1 ⇒ r < 1, hence all points inside Bloch sphere

represents mixed state. r = 0 represents a maximally mixed state.


Figure 1.3: Bloch Sphere

Quantum gates are unitary operations which can perform reversible computation.

Quantum gate operation transform a quantum state from one point of the Hilbert

space to another one. DiVicenzo in 1995 showed that a set of (i) general Single

Qubit Rotation (SQR) and (ii) a two qubit entangling gate forms a universal set

of gates for quantum computing.

Single Qubit Rotation (SQR) : A single qubit state can be visualised as a point

(θ, φ) on Bloch sphere (Fig. 1.24). Any trace preserving unitary operation move

this point over the surface and can be considered as a rotation of the state vector.

Using Pauli matrices,

σx =

[0 1

1 0

], σy =

[0 −ii 0

], σz =

[1 0

0 −1

], (1.26)

the rotation operators about x, y and z axes (Rx, Ry and Rz) can be written as,

Rx(θ) = e−iθ/2σx =

[Cos(θ/2) −iSin(θ/2)

−iSin(θ/2) Cos(θ/2)

],

Ry(θ) = e−iθ/2σy =

[Cos(θ/2) −Sin(θ/2)

Sin(θ/2) Cos(θ/2)

],

Rz(θ) = e−iθ/2σz =

[e−iθ/2 0

0 eiθ/2

].

(1.27)


Figure 1.4: Quantum circuit representation of Controlled-NOT gate (Eqn.1.29).

Important single qubit gates are,

NOT gate , UNOT = e−iπIx = −i

[0 1

1 0

]

Hadamard gate , UH = eiπ/2Iye−iπIx = −i

[0 1

1 0

]

Phase (φ) gate , Uφ = e−iφIz = e−iπ/2Iye−iφIxe−iπ/2Iy = e−iφ/2

[1 0

0 eiφ

] (1.28)

An operation on a system which is controlled by the state of another system is

called Controlled-Operators. In quantum computing these Controlled-Operators

are very important. A prototypical controlled operation is Controlled-NOT (Eqn.

1.29),

UCNOT =

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

(1.29)

When a quantum system in ground state is evolved under a slowly time varying

Hamiltonian, it stays in ground state of the instantaneous Hamiltonian. Adia-

batic quantum computation explores this idea for solving problems like finding

the ground state of a Hamiltonian, which is not easy to find. Recently molecular

ground state energy of Hydrogen is experimentally upto 45 bit accuracy by a two

spin NMR quantum computer [23].

Physical realizations of quantum computer

The basic qualities of a quantum system to be used as quantum information pro-

cessor are,


System Coherence time Gate operation timeNuclear spin 10−2 – 102 10−3 – 10−6

Electron spin 10−3 10−7

Ion Trap (In+) 10−1 10−14

Electron - Au 10−8 10−14

Electron - GaAs 10−10 10−13

Quantum dot 10−6 10−9

Optical cavity 10−5 10−14

Microwave cavity 100 10−4

Table 1.2: Different physical implementation of quantum computer and crudeestimates of coherence time and gate operation time in seconds (from [3]).

(i) Robust representation of quantum information as the states of the system,

(ii) Universal set of basic experimental unitary operations,

(iii) Sufficient coherence and gate operation time,

(iv) Ability to initialize the system quantum states for computation,

(v) Output measurement capability.

Many efforts are being made to realize a scalable quantum computer using the

techniques such as trapped ions, optical lattices, superconductor based qubits,

Bose-Einstein condensate based quantum computer [24] and NMR. First com-

mercial quantum computer was built by D WAVE (2011, in collaboration with

Lockheed Martin) based on superconducting flux qubits . In May 2013 Google

and NASA announced Quantum Artificial Intelligence lab with D WAVE’s 512

qubit quantum computer.

A comparison of coherence time and gate times of major candidates in experimen-

tal realization are given in Tab. 1.2 (from [3]).

1.2.1 NMR Quantum Information Processing

Long-lasting coherence and high fidelity controls in nuclear magnetic resonance

(NMR) are ideal for quantum information processing. NMR has become an

important experimental tool for demonstrating quantum algorithms, simulating

quantum systems, and for verifying various tenets of quantum mechanics. Experi-

mental implementation of quantum algorithms (Deutsch-Jozsa algorithm, Grover’s


search algorithm and Shor’s algorithm of factorization), testing basic principles of

quantum mechanics (no-hiding theorem [25], Leggett-Garg inequality [26]) and

quantum simulation (hydrogen molecule [23] and system with competing two and

three body interactions [27]) have been performed using Liquid state NMR. NMR

quantum computing uses the spin states of nuclei as qubits. The main limitations

of liquid state NMR quantum computer are,

(i) NMR quantum computation uses an ensemble of quantum systems (molecules),

typically of the size, N ≈ 1018. Thus it is an ensemble computing.

(ii) Creation of pure state is difficult in NMR. Pseudo Pure State (PPS), which

mimics the pure state is the initial state of most of the NMR quantum

information processor.

(iii) Scalability of NMR-QC depends on,

(a) availability of quantum spin system,

(b) robustness of the external controls and

(c) coherence time of the system.

1.2.1.1 Pseudo Pure State Creation

Equilibrium state in NMR is a statistical mixture of all the energy eigen states in

Boltzmann distribution. Cory et al. [28] showed an elegant method for creating

an effective pure state (or Pseudo Pure State - PPS) in NMR. PPS mimics the

pure state based on the idea: NMR experiments are sensitive only to traceless

deviation part of the density matrix.

In terms of product operators, the deviation density matrix (ρdev, Eqn. 1.23) of a

two spin system can be shown as (Eqn. 1.30),

ρdev = γ1I1z + γ2I

2z , (1.30)

where γ1 and γ2 are gyromagnetic ratios of first and second spins. For a homo-

nuclear spin system γ1 = γ2. The relative population of the eigen states |α α〉, |α β〉, |β α〉, |β β〉are 1, 0, 0,−1. Pseudo pure |α α〉 state is represented by,

ραα = I1z + I2

z + 2I1z I

2z . (1.31)


and the relative populations are 32,−1

2,−1

2,−1

2 and is equivalent to 2, 0, 0, 0

with an addition of identity matrix with proper weight factor (12). Other two spin

pseudo pure state density matrices are,

ραβ =− I1z + I2

z + 2I1z I

2z .

ρβα =I1z − I2

z + 2I1z I

2z .

ρββ =I1z + I2

z − 2I1z I

2z .

(1.32)

There are many methods for creating PPS in NMR. Most common methods are

(i) temporal averaging, (ii) spatial averaging [28], (iii) logical labelling [29] and

(iv) SALLT [30].

Temporal averaging (TA) uses a set of states with average state behaves like pure

state. TA does not really create a logical pure state, but average result can simulate

the execution of a logical operation.

Spatial Averaging (SA) is the most commonly used method for PPS creation

in NMR [28]. SA is based on dividing the system in spatially averaged sub-

ensembles. These sub-ensembles can be manipulated independently using rf pulses

and gradients. We have used SA method for creating PPS in quantum simulation

experiments described in chapters 2 and 4.

Logical or state labeling was first introduced by Gershenfeld and Chuang [29]. In

this method, for an n qubit system, one qubit is used for label the state, while

the (n − 1) qubits are put in a PPS configuration. Spatially Averaged Logical

Labeling Technique (SALLT) [30], combines spatial averaging and logical labeling

techniques. This method uses one labeling qubit and pseudopure state of any

number of qubits can be created in the subsystem manifolds of the labeling qubit.

SALLT does not scale with number of qubits.

1.2.1.2 Quantum State Tomography (QST)

Readout in NMR is performed by recording the single quantum spectra – equiv-

alent to measuring x and y component of the magnetization or the expectation

value 〈Ix + iIy〉 – which is a function of the final observed density matrix (ρ).

Density matrix - spectrum is a many to one mapping and needs around 2n × 2n

spectra for exact density matrix reconstruction (or quantum state tomography).

QST is a systematic way to measure all the elements of the density matrix ρ. The


first NMR tomography was developed by Chuang et al. [31].

Consider a general deviation density matrix of a coupled two spin system,

∆ρ =

x11 x12 + iy12 x13 + iy13 x14 + iy14

x12 − iy12 x22 x23 + iy23 x24 + iy24

x13 − iy13 x23 − iy23 x33 x34 + iy34

x14 − iy14 x24 − iy24 x34 − iy34 x44

(1.33)

∆ρ is a traceless Hermitian ⇒ x11 + x22 + x33 + x44 = 0.

Single quantum coherences of ∆ρ (Eqn. 1.33) for spin I1 and I2 are (x12 +

iy12), (x34 + iy34) and (x13 + iy13), (x24 + iy24) respectively. These ∆ρ12, ∆ρ34,

∆ρ13 and ∆ρ24 are the ‘reading positions’ of the density matrix. By unitary

transformations, any desired element of density matrix can be brought in to these

‘reading positions’. Explicit solution for two spin NMR system is given in [32].

1.3 Genetic Algorithm optimization

“The gene is by far the most sophisticated program around.”

- Bill Gates, Business Week, June 27, 1994.

Optimization is a mathematical procedure of finding the best element from a set

of available alternatives and is central to any problem involving decision making.

Finding optimal parameter values or conditions for most favorable performance

of a function makes it popular in all areas of science and technology. Aided by

rapid growth in computer science, optimization has received enormous attention

in recent years.

The basic optimization problem can be stated as,

minimize or maximize f(x)

subject to x ∈ Ω(1.34)


The function f : Rn −→ R, is a real values function and is called the objective

function or cost function. ‘x’ is a vector of ‘n’ independent variables referred to

as ‘decision variable’. The set Ω is a subset of Rn. Out of all possible x (∈ Ω)

vectors, optimization algorithm has to find the ‘best’ x (say xb) vector which will

give the minimal or maximal value – f(xb) – for the function. As an example,

function profile of g(x1, x2) (Eqn. 1.35) – known as ‘peak ’ in Matlab, obtained by

translating and scaling Gaussian distributions – is shown in Fig. 1.5.

g(x1, x2) = 3(1− x1)2 × exp[−x21 − (x2 + 1)2]− 10(x1/5− x3

1 − x52)

×exp[−x21 − x2

2]− 1

3× exp[−(x1 + 1)2 − x2

2](1.35)

In the two dimensional space defined by the vectors x1, x2, function g has many

local maximas (mx1, mx2 and mx3) and minimas (mn1). An optimization algo-

rithm with defined problem, maximize g(x), will search this two dimensional space

for the global maxima point mx1.

Figure 1.5: Function profile of g(x1, x2) (Eqn. 1.35). Maximas (mx1, mx2

and mx3) and minimas (mn1) of the function are marked.

Genetic algorithms (GA) are stochastic global search method [33] based on the

mechanics of natural biological evolution (Fig. 1.6). It was proposed by John

Holland in 1975[34]. GA operates on a population of solutions of a specific prob-

lem by encoding the solutions to a simple chromosome-like data structure, and

applies recombination operators. At each generation, a new population is created

by breeding individuals (selected according to their fitness value) together using

operators borrowed from natural genetics. This process leads to the evolution of

individuals and generate populations that are better suited to their environment.

A comparison of Darwinian evolution and GA evolution is given in Tab. 1.3.

GAs differs substantially from other traditional optimization methods, which are,


Darwinian Evolution Genetic AlgorithmEnvironment. −→ Optimization problem.Individuals living in that envi-ronment.

−→ Feasible solutions.

Individual’s degree of adaptationto its surrounding environment.

−→ quality of solutions or fitnessfunction.

A population of organisms(species).

−→ A set of feasible solutions.

Selection, recombination andmutation in natures evolutionaryprocess.

−→ Stochastic operators.

Evolution of populations to suittheir environment.

−→ Iteratively applying a set ofstochastic operators on a set offeasible solutions.

Table 1.3: Darwinian Evolution and GA optimization problem comparison.

Figure 1.6: Mechanism of biological evolution.

(i) Traditional methods generate a single point at each iteration while GA search

a population in parallel.

(ii) Selects the next population by probabilistic transition rules (by using random

number generators).

(iii) Do not require any auxiliary information about the problem (like derivatives

in gradient ascent algorithm).

GAs are attractive in engineering design and applications because they are easy

to use and are likely to find the globally best design or solution, which is superior

to any other design or solution [35]. Genetic Algorithm optimization has found

applications in various fields of physics such as molecular geometry optimization

[36], prediction of ultrahigh-pressure phases of ice [37], optimization of silicon clus-

ters [38], and determination of best-fit potential parameters for a reactive force

field [39]. Genetic algorithm has been used in NMR for designing new experiments

[40], for improving excitation and inversion accuracy of rf pulses [41] and pulse


sequence optimization [42].

The first step in GA optimization is defining a good genetic representation that

is expressive and evolvable. Encoding solutions of the problem to individuals in

genetic evolution is performed using this representation scheme and optimality

of these solutions or individuals are defined by fitness function. Flow of genetic

evolution process of the population is mainly controlled by fitness function and is

crucial for any GA optimization. There can be many fitness function for the same

problem and selection is usually based on sensitivity of the function – change in

fitness with change in generated solutions – and computational complexity.

In our work we have explained representation scheme for designing pulse sequence

and for delay optimization in quantitative INEPT . An objective function that

prescribes the optimality of a solution or individual in this representation space is

called fitness function.

Outline of the basic Genetic Algorithm optimization is (Fig. 1.7) given by,

(i) The optimization begins by creating a random initial population of individ-

uals (or valid solutions to the problem).

(ii) Creates a sequence of new populations based on following steps:

(a) Evaluate the fitness of each member of the current population.

(b) Select two Parent individuals from the population according to their

fitness (the better fitness, the bigger chance to be selected).

(c) Some of the individuals in the current population that have lower fit-

ness are chosen as elite. These elite individuals are passed to the next

population (Fig. 1.9).

(d) Produces Children from the Parents. Children are produced either by

making random changes to a single Parent – mutation – or by combining

the vector entries of a pair of Parents – crossover.

(e) Replaces the current population with the Children to form the next

generation.

The algorithm stops when one of the stopping criteria is met.


Figure 1.7: Basic outline of Genetic Algorithm.

1.3.1 Population Initialization

Random generation of the individual parameters, which follows the specific con-

straint levels of the problem, is the most common way of creating initial population

in GA optimization. Prior knowledge about the solution properties can be utilized

for creating a fast converging initial population. The size of the population has

to be decided by the user based on the complexity of the problem and available

computational power.

1.3.2 Selection Operators

Selection options specify how the genetic algorithm chooses Parents for the next

generation. Common selection methods are random selection (each individuals

have equal probability for getting selected without any reference to its fitness

value), proportional selection (based on the fitness value of the individual) and

rank-based selection (based on the rank order of the fitness value).

The specific proportional selection functions we have used for optimization are,

Stochastic uniform: This method lays out a line in which each Parent corresponds

to a section of the line of length proportional to its scaled value. The algorithm

moves along the line in steps of equal size. At each step, the algorithm allocates

a Parent from the section it lands on. The first step is a uniform random number

less than the step size.

Roulette: Roulette selection chooses Parents by simulating a roulette wheel, in

which the area of the section of the wheel corresponding to an individual is pro-

portional to the individual’s expectation. The algorithm uses a random number


Figure 1.8: Roulette Wheel selection method for a popuation of six individualsA, B, C, D, E, F. The area for each individual is proportional to its fitnessvalue.

Figure 1.9: Three different types of Children from existing generation to newgeneration.

to select one of the sections with a probability equal to its area.

1.3.3 Reproduction Operators

The selected individuals or Parents are combined, using reproduction operators.

These operators are borrowed from natural genetics to generate new generation.

The reproduction operators are designed in such a way that the favorable genetic

material should be preserved throughout generations and at the same time pop-

ulation must have genetic diversity. Children are always feasible with respect to

specific constraints of the problem. One or more most fit individuals are allowed

to propagate through successive generations known as elite Children. This elistic

strategy will make sure an incremental or steady state evolution. The most com-

mon reproduction operators are crossover and mutation (Fig. 1.9).

1.3.3.1 Crossover Operators:

Crossover specifies how pairs of Parents should be combined to form a Child for

the next generation (Fig. 1.9). Crossover function randomly selects an entry of


the Child vector and randomly replace with same coordinate entry from one of

the two Parents. The crossover operators used in our optimisation are,

(i) Single point : randomly select one point (n) in the Child vector and replace

the entries before (1 to n) and after (> n + 1) the point from Parent1 and

Parent2 respectively (Fig. 1.10(a)).

(ii) Two point : selects two random points (m and n) in the Child vector. The

two point crossover operator replace the entries of the Child vector 1 to m

and n + 1 to final from Parent1 and remaining entries from Parent2 (Fig.

1.10(b)).

(iii) Scattered : based on a randomly generated entries of a binary array (0 or 1),

entries of the Child vector is selected from Parent1 (if binary element is 0)

or Parent2 (Fig. 1.10(c)).

(iv) Intermediate: Child creation is based on weighted average of the Parents.

The parameter ‘Ratio’ can be used to control the ‘Child ’.

Child = Parent1 + rand×Ratio× (Parent2− Parent1). (1.36)

where rand is a random number.

(v) Heuristic: Heuristic crossover operator creates Child from Parent1 and

Parent2 based on the following formula,

Child = Parent2 +Ratio× (Parent1− Parent2), (1.37)

which creates a Child that lies on the line containing the two Parents, closer

to the Parent with the better fitness value. The ‘Ratio’ can be used to con-

trol the Child generation.

(vi) Arithmetic: creates Child that are the weighted arithmetic mean of Parent1

and Parent2.

1.3.3.2 Mutation Operators

Mutation Operators randomly change the genes of individual Parents and intro-

duce new genetic material. Mutation provides genetic diversity which enable the


(a) Single Point Crossover

(b) Two Point Crossover

(c) Scattered Crossover

Figure 1.10: Different Crossover operators.

genetic algorithm to search a broader space. Mutation is usually applied with low

probability to preserve favorable genetic material. The mutation operators used

in our optimisation are,

(i) Gaussian: Adds a random number taken from a Gaussian distribution with

mean 0 to each entry of the Parent vector. The standard deviation of this

distribution is determined by the parameters Scale and Shrink,

(ii) Uniform: Selects a fraction of the vector entries of an individual for mu-

tation, where each entry has a probability rate of being mutated and the

algorithm replaces each selected entry by a random number selected uni-

formly from the range for that entry.

(iii) Adaptive Feasible: Randomly generates directions that are adaptive with

respect to the last successful or unsuccessful generation.

Stopping conditions of GA optimization is mainly implemented using fitness limit

or optimization time or generations or function tolerance. We have used GA

optimisation for,

(i) pulse sequence generation for a given operator / quantum gate,

(ii) quantum state generation (with specified initial and final state),

(iii) generation of broadband and rf composite pulses and

(iv) delay set optimisation for quantitative INEPT (GAQIC). The GA optimi-

sation details such as representation scheme, fitness function, mutation op-

erators, crossover operators, population size, number of generations etc are

different in each case and are explained with details in respective chapters.


All the programs are written in Matlab®-7.14 in combination with Matlab’s opti-

mization toolbox. All optimizations are performed in a 3.6 GHz Intel® Core i5

Processor processor with Ubuntu 12.04 operating system (32 bit).

Chapter 2

Genetic Algorithm Optimization

in NMR Quantum Information

Processing

“Hilbert space is a big place”

- Carlton Caves.

2.1 Introduction

Decomposing a unitary operator as a sequence of experimentally preferable oper-

ators is the main task in experimental implementation of a quantum algorithm.

There are several proposals for such decomposition in NMR QIP, such as SMPs

(Strongly Modulated Pulses) by Fortunato et al. [43], GRAPE (Gradient Ascent

Pulse Engineering) by Khaneja et al. [44], and algorithmic approach by Ajoy et

al. [45]. Here we investigate the use of Genetic Algorithm for direct numerical

optimization of pulse sequences, and devise a probabilistic method for perform-

ing universal quantum computation using hard pulses. GA optimization, being a

global search algorithm, yields unitary decompositions that are more general, and

hence can be applied to any spin system with different values of J couplings and

chemical shifts. We also investigate quantum state preparation using GA opti-

mization. For performing non-unitary transformations, we have included Pulsed

23

Chapter 2. Genetic Algorithm Optimization in NMR Computing 24

Field Gradients (PFG) [46] in GA optimization. We demonstrate hard pulse uni-

tary decomposition for preparation of Pseudo-Pure State (PPS) and Long Lived

Singlet State (LLSS) (along with other three Bell states) directly from thermal

equilibrium in a two qubit homonulear NMR system. Section 2.2 describes the

optimization procedure and Sec. 2.3 outlines the experimental implementations.

2.2 Genetic Algorithm for NMR Pulse Sequence

Generation

In liquid state NMR, the system Hamiltonian is composed of the interactions of

nuclear spins with the external magnetic field and with each other. Combining

that with external radio frequency (RF) pulses (with specific frequency, amplitude

and phase), one can simulate any preferred effective Hamiltonian [14]. Hence the

unitary operator decomposition problem in NMR can be treated as an optimiza-

tion problem that gives optimal values of pulse parameters and delay durations.

Optimality is determined here by a proper fitness function, which depends on the

target Hamiltonian or the target state.

We have performed pulse sequence optimization using GA for quantum logic gates

(operator optimization) and quantum state preparation (state-to-state optimiza-

tion) [43]. State-to-state optimization converges faster than operator optimization

(there can be many operators which can perform the same state-to-state transfer).

In the discussion given below, Single Qubit Rotation (SQR) pulses and two-qubit

Controlled-NOT (CNOT) gates are operator optimizations, whereas creation of

PPS and Bell states are state-to-state optimizations.

2.2.1 Representation Scheme

Representation scheme is the method used for encoding the solution of the problem

in individuals undergoing genetic evolution. We have selected the gene as a combi-

nation of (i) an array of pulses that we apply simultaneously on each channel with

arbitrary amplitudes (θ) and phases (φ), and (ii) arbitrary delays (d) between the

pulses. It can be easily shown that repeated application of the above gene forms

the most general pulse sequence in NMR. Let an individual representing a valid


solution have m genes, and let n be the number of channels or spins. Then the

individual can be described as a matrix of size (n+ 1)× 2m as shown in Eq. 2.1.

θ11 φ11 . . θm1 φm1

θ12 φ12 . . θm2 φm2

. . . . . .

. . . . . .

θ1n φ1n . . θmn φmn

d1 0 . . dm 0

(2.1)

This matrix has to be optimized, according to the optimality condition expressed

by the fitness function (Sec. 2.2.2).

To begin with, we guess the number of genes (m) as a number that depends on

the complexity of the problem to be solved. If the fidelity, i.e. the fitness of the

best individual in the population, calculated using the fitness function crosses a

cutoff value of (say > 99%), the optimization program tries to reduce the number

of genes by assigning zero values to a few gene parameters. If the fitness function

does not cross the cutoff value, the program will re-run with more number of

genes. We have carried out the optimization procedure using a population size

of 100 individuals evolving for 1000 generations. All the programs are written in

Matlab® in combination with Matlab’s optimization toolbox.

2.2.2 Fitness Function

A fitness function is a particular type of objective function that describes the op-

timality of a solution or individual. In operator optimization, GA tries to reach

a preferred target unitary operator (Utar) from an initial random guess pulse se-

quence operator (Upul). We have selected the fitness function Fpul to be the pro-

jection of Upul onto Utar,

Fpul = Trace(Upul × U †tar). (2.2)

It is normalized to give the maximum value 1.0 when Upul=Utar.

In state-to-state optimization, the optimization program will run over different

possibilities of Upul, to prepare a preferred target state ρtar from the initial state


Figure 2.1: A two-spin system with chemical shifts ±δ and coupling J

ρin. Then we choose the fitness function to be

Fstate = Trace(Upul × ρin × U−1pul × ρ

†tar). (2.3)

In both cases, the optimization has to maximize the fitness function.

2.3 Two qubit homonuclear case

Consider the two qubit NMR homonuclear system (Fig. 2.1) with chemical shifts

±δ and coupling J . Assuming weak coupling (δ J), the Hamiltonian can be

written as [14],

H = Hcs +HJ = 2πδ(Iz1 − Iz2) + 2πJ(Iz1Iz2). (2.4)

For single qubit rotations in this system, one can use spin selective pulses (low

power, long RF pulses) which will excite a small spectral region around the selected

spin [47]. On the other hand, by using the natural chemical shift difference between

two spins, we show here how to implement SQR with global hard (non-selective)

pulses. Later we extend this method to perform two qubit homonuclear universal

quantum computation using only global hard pulses.

2.3.1 Operator Optimization

Operator optimization deals with pulse sequence generation for quantum logic

gates. Here we look at two essential unitary operators for universal quantum

computation: SQR and the two-qubit CNOT gate.


Figure 2.2: Pulse sequence for single qubit rotation. First two filled pulsesare (π/2). Flip angle of the third pulse is (θ/2) with phase φ. The parenthesesabove each pulse contain the flip angle (first number) and the phase (secondnumber).

2.3.1.1 Single Qubit Rotations using non-selective pulses

For a two qubit homonuclear NMR system, we first consider the case J = 0,

H = Hcs = 2πδ(Iz1 − Iz2). (2.5)

Evolution under such a Hamiltonian creates a relative phase among spins, pro-

portional to the chemical shift difference (2δ) and the evolution time. Combining

this relative phase with global rotation hard pulses, single qubit operations can be

performed.

The target operator for SQR is,

Utar = exp(−iθIφk), (2.6)

where k=1 or 2, θ is the flip angle and φ is the phase. The optimized pulse

sequence for SQR is shown in Fig. 2.2. At the start, we selected m = 3, i.e. three

hard pulses and three delays. The optimized sequence has three hard pulses and

a single delay. The flip angle (θ) of the SQR pulse is determined by the delay and

the flip angle of the third pulse, whereas the phase of SQR (φ) and spin selection

is determined by phases of all the three pulses (Tab. 2.1). It may be pointed out

that single qubit rotation with θ = π is a NOT gate and θ = π/2 is a pseudo

Hadamard gate [30].

Experimental verification of the optimized SQR pulse sequence in 5-Bromofuroic

acid (Fig. 2.3(a)) (in C6D6) is shown in Figs. 2.3(c) and 2.3(d). The total length

of the pulse sequence for the (π/2) SQR pulse is less than 500 µs, whereas the

conventional method (using a selective soft pulse) would need a 2 ms shaped pulse.


Spin to be Excited φ1 φ2

1 (φ− π/2) (φ+ π/2)2 (φ+ π/2) (φ− π/2)

Table 2.1: Spin selection and phase φ of the SQR is controlled by the phasesφ1 and φ2.

This substantial shortening in time can lead to a significant advantage in quantum

circuits.

The above analysis also holds for J 6= 0, so long as γB1 δ, J , except that

introduction of the J coupling dephases the final state and results in a fidelity loss.

The fidelity of the pulse sequence (fitness function Eq. 2.2) is studied using Matlab

simulation (Fig. 2.4) and is > 99.8% for J/δ < 0.1 and θ < π/2. (J/δ < 0.1 is the

limit for weakly coupled spins and here we are dealing with only weakly coupled

spins). For π/2 < θ < π, the theoretical fidelity is still quite good (>99.5%) as

can be seen in Fig. 2.4.

2.3.1.2 Controlled NOT gate

The CNOT gate is an essential component in the construction of a universal quan-

tum computer. Any quantum circuit can be simulated to an arbitrary degree of

precision using a combination of CNOT gates and single qubit rotations [3]. The

target unitary operator for C1NOT2 (control on first qubit and NOT on second

qubit) is

Utar = exp(−i(−π4I +

π

2Ix2 +

π

2Iz1 − πIz1Ix2)). (2.7)

The optimized pulse sequence for CNOT is shown in Fig. 2.5(a), obtained using

the Hamiltonian of Eq. 2.5. All the four CNOT gates can be obtained by tuning

(θ, φ) as shown in Tab. 2.2. The pulse sequence is identical for all the four CNOT

gates except for the angles θ and φ.

The experimental implementation of various CNOT gates in 5-Bromofuroic acid

is illustrated in Fig. 2.5(b). We achieved an average experimental fidelity greater

than 99%.

The theoretical fidelity of the CNOT operation (as per Eq. 2.2) is dependent on

the ratio (J/δ). Matlab simulation of fidelity as a function of (J/δ) for the CNOT

gate pulse sequence is shown in Fig. 2.5(c). We find that fidelity is > 99.99% for


(a)

(b)

(c)

(d)

Figure 2.3: (a) Chemical structure of 5-Bromofuroic acid. Diagonal elementsin the table contain the chemical shifts of protons at 500 MHz and the non-diagonal element represents the J coupling (sample dissolved in C6D6). (b)Equilibrium spectrum. (c) (π/2)y SQR pulse on spin-1. (d) (π/2)y SQR pulseon spin-2. The average experimental fidelity (calculated using the standarddefinition [43] that compares spectral intensities with the equilibrium spectrum)for the SQR pulse is greater than 99%.

Gate θ φC1NOT2 π/4 π/2C1NOT2 π/4 0C2NOT1 3π/4 0C2NOT1 3π/4 π/2

Table 2.2: (θ,φ) values for all four CNOT gates. The notation used is CiNOTj ,denoting control qubit i and target qubit j. Also Ci means that NOT operationacts on the target when the control qubit is in the |0〉 state.


Figure 2.4: Matlab simulation study of fidelity variation of SQR with differentvalues of J/δ and θ. The special case for 5-Bromofuroic acid (J/δ = 0.014) isshown with an arrow in x axis.

(J/δ) = 0.01 and drops to 99.84% for (J/δ) = 0.1. This means that even if one

needs ten CNOT gates in a quantum circuit, the fidelity can exceed 99%.

2.3.2 State to State Optimization

State to state optimization deals with pulse sequence generation for quantum state

preparation. Here, we have added gradient pulses to the optimization procedure,

which enabled us to perform non-unitary transformations. We performed two

important quantum state preparations: pseudo-pure state creation, and Bell state

creation directly from the mixed thermal equilibrium state, using global hard

pulses.

2.3.2.1 Pseudo-Pure State Creation

Quantum information processing by NMR spectroscopy uses PPS to mimic the

evolution and observations on pure states [28]. There are several methods for

creating PPS from thermal equilibrium [28–30]. We closely follow the Spatial

Averaging Method (SAM) of Cory et al.. SAM uses qubit selective pulses, which

in homonuclear qubit systems become soft long pulses. Instead, we obtain a novel

pulse sequence using only non-selective (hard) pulses for a homonuclear two qubit

system. The optimization problem has the thermal equilibrium state ∆ρeq = Iz1 +

Iz2 as the initial state and the ∆ρ00 = Iz1 +Iz2 +2Iz1Iz2 as the target state. SAM

also uses pulse field gradient (PFG) pulses to defocus all transverse magnetization

components and retain only longitudinal components of magnetization. These are


(a)

(b)

(c)

Figure 2.5: (a) Pulse sequence for CNOT gate. Parentheses above each pulsecontain flip angle (first number) and phase (second number). (b) Diagonalelement tomography of: (i) equilibrium state, and states obtained after applying(ii) C1NOT2, (iii) C2NOT1, (iv) C1NOT2, (v) C2NOT1. The labels 1, 2, 3,4 represent the states |00〉, |01〉, |10〉, |11〉. An average experimental fidelitygreater than 99% is observed (calculated according to Ref. [43] from diagonalelements of the density matrix). (c) Theoretical fidelity (F ) vs. (J/δ) plot forCNOT gate.


Bell State φ1 φ2 φ3 d1 d2

|ψ+〉 = 1√2(|00〉+ |11〉) 3π/4 9π/8 3π/4 1/16δ 0

|ψ−〉 = 1√2(|00〉 − |11〉) 3π/4 9π/8 π/4 1/16δ 0

|φ+〉 = 1√2(|01〉+ |10〉) 0 5π/8 3π/4 9/48δ 9/8δ

|φ−〉 = 1√2(|01〉 − |10〉) 0 5π/8 π/4 9/48δ 9/8δ

Table 2.3: Values of φ and d for Bell state preparation using the pulse sequenceshown in Fig. 2.7(a).

the non unitary operations needed to convert a mixed state to a pure/pseudo pure

state.

For easier optimization and experimental implementation, we fixed all the pulses to

be (π/2) and optimized only the delays and pulse phases. The resulting sequence

consists of six (π/2) pulses and one (π) pulse for refocusing the chemical shift (Fig.

2.6(a)). The phase of the (π) pulse can be controlled to achieve either the |00〉PPS or the |11〉 PPS. The other PPS are obtained using a combination sequence

of PPS and a SQR π pulse. The experimental results are shown Fig. 2.6(b). An

average experimental fidelity greater than 99% is obtained for various PPS.

The theoretical fidelity (using Eq. 2.3) of the PPS preparation pulse sequence is

also dependent on the ratio J/δ. Matlab simulation of this fidelity as a function

of J/δ is shown in Fig. 2.6(c). We observe that fidelity is > 99.9% for J/δ < 0.1.

2.3.2.2 Creation of Bell States Directly from the Thermal State

Bell states are maximally entangled two-qubit states (also known as the Einstein-

Podolsky-Rosen states) [48]. They play a crucial role in several applications of

quantum information theory. They have been used for teleportation, dense coding

and entanglement swapping [49–51]. Creation of Bell states using NMR conven-

tionally requires PPS creation + Hadamard gate + CNOT gate, and hence is

demanding [52]. We integrated all these steps in a single pulse sequence, and op-

timized that with GA. Again, we kept all pulse amplitudes to be hard (π/2) and

optimized the pulse phases, delay durations and positioning of PFG pulses. The

optimized pulse sequence (Fig. 2.7(a)) has ten non-selective pulses. The final Bell

state can be selected by controlling the phase of the pulses and delay durations

according to Table 2.3.


(a)

(b)

(c)

Figure 2.6: (a) Pulse sequence for PPS creation. All filled pulses are (π/2),and non-filled one is (π), with phases written above them. ± in phase of (π)pulse determines PPS to be created (|00〉 or |11〉). Shaped pulse along Gz rep-resents a PFG pulse, which defocuses all transverse magnetization componentswhile retaining longitudinal magnetization components [46]. (b) Tomography ofdiagonal elements after preparing (i) |00〉 PPS, (ii) |01〉 PPS, (iii) |10〉, and (iv)|11〉 PPS. The labels 1, 2, 3, 4 represent the states |00〉, |01〉, |10〉, |11〉. Anaverage experimental fidelity greater than 99% is observed (calculated accordingto Ref. [43] from diagonal elements of density matrix). (c) Theoretical fidelity(F ) vs. (J/δ) plot of PPS generation pulse sequence.


(a)

(b) (c)

Figure 2.7: (a) Pulse sequence for creating the Bell states directly form thethermal equilibrium state. All filled pulses are (π/2) and non-filled pulses are(π), with the phases written above them. The shaped pulse along Gz representsa PFG pulse which defocuses all the transverse magnetization components, re-taining longitudinal magnetization components [46]. The values of φ’s and ‘d’sare listed in Table 2.3. (b) Observation of the singlet state after applying theoperator U of Eq. 2.9. (c) Density matrix tomography of the created singletstate . The labels 1, 2, 3, 4 represent the states |00〉, |01〉, |10〉, |11〉.

Experimental preparation of the Bell state |φ−〉 = 1√2(|01〉 − |10〉), also known as

the long-lived singlet state (LLSS), is carried out in 5-Bromofuroic acid with the

Hamiltonian H = I1 · I2 [53, 54]:

ρφ− = (0.25I − Ix1Ix2 − Iy1Iy2 − Iz1Iz2). (2.8)

The experimental results are shown in Fig. 2.7(b) and 2.7(c). Since ρφ− is not

directly observable, we convert the created singlet state into observable single

quantum coherence by applying

U = e−i(π/2)Ix1 × e−i(π/2)Ix2 × e−i(π/4)(Iz1−Iz2). (2.9)

An experimental fidelity of more than of 99% is achieved. The pulse sequence

given in Fig. 2.7(a) is the shortest known pulse sequence for creating pure singlet

state in a two-qubit homonuclear NMR system [55].

The singlet state life time (Ts) was measured by applying WALTZ-16 spin-lock

sequence [56] for a variable time period (0− 20 s). We find it to be 11.2 s (Fig.

2.8), which is longer than T1 = 8.7 s and T2 = 3.8 s. Mahesh et al. have found Ts


Figure 2.8: Anti-phase signal decay as a function of interval and fits to asingle exponential decay. The initial intensity of singlet state is normalizedto one. Observed Singlet state life time is Ts = 11.2 Sec. The system has aT1 = 8.7 Sec and T2 = 3.8 Sec.

is rather sensitive to decoupling sequence used [57] and hence Ts can be improved

with different decoupling sequences.

The SQR and CNOT gates implemented using hard pulses (Sections 2.3.1.1 and

2.3.1.2) are valid for larger qubit systems with homonuclear spin pairs (for example

the 1H − 1H − 19F − 19F system in 2,3-Difluro-6-nitrophenol [47]).

2.4 Conclusion

We have used the global optimization power of Genetic Algorithm for,

(i) efficiently implementing SQR and CNOT gates, and

(ii) creating PPS in homonuclear two-qubit system using only hard pulses.

That demonstrates a method for performing universal quantum computation in

such systems. We also demonstrated the creation of LLSS and Bell states directly

from thermal equilibrium state, with the shortest known pulse sequence. It should

be noted that all the pulse sequences discussed here are generic, independent of

the system and the spectrometer.

Chapter 3

Quantum Simulation Using

Fidelity Profile Optimization

“We can’t solve problems by using the same kind of thinking we used when we

created them”

- Albert Einstein

3.1 Introduction

Algorithms with exponential speedups over classical counterparts [58, 59], simu-

lation of quantum systems [23, 60, 61] and testing basic principles of quantum

mechanics [25, 26] makes quantum Information processing (QIP) and quantum

computation intensively investigated fields of physics over last decade. The idea

of simulating quantum systems in a quantum computer was proposed by Feynman

[62] in 1982 and is one of the most important practical application of the quan-

tum computer. Quantum simulation has the potential to revolutionize physics and

chemistry as demonstrated recently by solving problems like – molecular Hydrogen

simulation [23, 63, 64], calculations of thermal rate constants of chemical reactions

[61] and quantum chemical dynamics [65].

37

Chapter 3. Quantum Simulation using Fidelity Profile Optimization 38

Experimental implementation of a quantum unitary operation / gate (U) requires

decomposition of U in terms of experimental system unitaries (such as exter-

nal control operators and natural Hamiltonian evolution of the experimental sys-

tem). We have explained this Unitary Operator Decomposition in previous chap-

ter (Chapter 2) for NMR quantum information processor. Henceforth we will

use the abbreviation UOD for Unitary Operator Decomposition. Fidelity Pro-

file Optimization (FPO) is a generic version of UOD with variable target unitary

operator. Consider a Hamiltonian, H(x1, x2...xn), with time evolution operator,

U(x1, x2...xn, t). FPO is a systematic way of evaluating the most generic sequence

of experimental unitary operators with x1, x2...xn, t dependence. We classify

FPO on the basis of number of independent variables in the functional representa-

tion of the unitary operator (‘nth order FPO’ – for n independent variables). The

UOD optimization for constant target unitaries (zero independent variable) such

as C-NOT, CC-NOT, C-Hadamard etc are zeroth order FPO, which will give a

point in fidelity space (spanned over the range of Hamiltonian parameters). One

independent variable will give a line in the fidelity space, while a two indepen-

dent variable optimization will give a plane. Thus the order of FPO decides the

dimension of fidelity space.

Long-lasting coherences and high fidelity controls in nuclear magnetic resonance

(NMR) are ideal for quantum information processing. Experimental implementa-

tion of quantum algorithms (Deutsch-Jozsa algorithm, Grover’s search algorithm

and Shor’s algorithm of factorization ), testing basic principles of quantum me-

chanics (no-hiding theorem [25], Leggett-Garg inequality [26]) and quantum sim-

ulation (hydrogen molecule [23] and system with competing two and three body

interactions [27]) have been performed using Liquid state NMR (LS-NMR). LS-

NMR with weak coupling approximation has ‘zz’ interaction between spins (Eqn.

3.1),

Hzz = Jzz(σ1zσ2z), (3.1)

(where Jzz is the interaction strength). Hzz with single qubit rotations (SQR)

Rn(θ, φ) (θ angle along φ axis on nth spin),

Rn(θ, φ) = exp(−iθ/2× [Cosφ σnx + Sinφ σny]), (3.2)

forms universal set of unitary operators for quantum information processing.

Chapter 2, describes the use of GA for optimal design of various one and two qubit


quantum gates by NMR. Genetic algorithm has been used in NMR for designing

new experiments [40], for improving excitation and inversion accuracy of RF pulses

[41] and pulse sequence optimization [42, 66]. Here we use Genetic algorithm

optimization for solving UOD for generic DM Hamiltonian with Heisenberg-XY

interaction.

Section 3.2 deals with theoretical discussion of FPO followed by experimental

implementation in Section 3.3.

3.2 Theory - Fidelity Profile Optimization (FPO)

Fidelity Profile Optimization is performed by a systematic evaluation of Hamilto-

nian parameter dependence in the decomposed experimental unitaries. We have

used multiple optimizations using Genetic Algorithm [33], where complexity of

optimization depends on Hilbert space dimension as well as order of the FPO.

Zeroth order FPOs – Controlled NOT and Hadamard gates – were explained in

Chapter 2. The details of the specific FPO - GA optimization used here are,

Representation Scheme: A valid UOD is represented as a matrix of size (n+1)×2m

(given in Chapter 2). Here ‘n’ is the number of qubits and ‘m’ varies with the

complexity of the operator. So the problem is to find an optimized matrix where

optimality condition is posed by a fitness function.

Fitness Function: We have selected the fitness function Fuod to be the projection

of Uuod onto Utar,

Fuod = Tr(Uuod × U †tar). (3.3)

GA optimization procedure of a first order FPO, Controlled-φ gate (Eqn. 3.7), is

explained in following steps as an illustrative example.

Step.i : Perform zeroth order FPO of Controlled-φ gate for a typical value of φ

(say φ = π/2) (as explained in Chapter 2).

Step.ii : Starting from the individual with maximum fidelity, from Step.i, we will

identify the φ dependent UOD parameters (individual array elements –

amplitudes, phases and delays) in this step. This is accomplished by iter-

atively checking φ dependence of each UOD parameters. For Controlled-φ

gate, there are three φ dependent UOD parameters.


Step.iii : Identify functional dependence of UOD parameters, by substituting each

UOD parameters by basic functions such as polynomials, exponential

functions, Fourier functions etc. and solve for the functional constants by

performing GA optimization again. Complexity of functional dependence

vary systematically in such a way that functions with minimum number

constants can be obtained.

For Controlled-φ gate, substituting each UOD parameters with degree one poly-

nomial, aiφ+bi and performing GA optimization, constants (ai, bi) are obtained

as (1/2, 0), (1/4, π/2), (-1/4, π).

Second order FPO is also performed by the same procedure described above by

independently checking each FPO parameters for its Hamiltonian variable depen-

dency. Nature of the functional dependence (single variable or two variable) of

FPO parameters is also important in this case. Single variable dependent UOD

parameters can be identified by substituting basic functions as described above for

Controlled-φ case. Two variable dependent UOD parameters functions are iden-

tified by substituting product of basic functions. A direct generalization to this is

possible for n variable dependent unitary operators (or nth order FPO).

All FPOs discussed here are based on basic two qubit interaction as Hsys = Hzz

(weak coupling in nuclear magnetic resonance; (Eqn. 3.1) with evolution unitary,

Uzz(θ) = exp(−i θ/2 σzσz). (3.4)

However it can be generalized to almost any system interaction by term isolation

procedure by Bremner et al. [67] and Hill et al. [68]. As an example consider the

case,

Hsys = J(σxσx + σyσy + σzσz). (3.5)

The Hzz terms can be isolated from Eqn. 3.5 by (as shown in [68]),

exp(−iJzzσzσzt) = R1(π, z)exp(−iHsyst)R1(π, z)

exp(−iHsyst).(3.6)

All the programs are written in Matlab in combination with Matlab’s optimization

toolbox. The typical time of FPO varies exponentially with number of qubits (since

Hilbert space dimension grows exponentially). A typical first order FPO for three


spin system (CC-φ gate in Sec. 3.2.1.2) took nearly three hours on a computer

with 3.6 GHz processor (Matlab® 7.14; 32 bit Ubuntu 12.04 OS).

3.2.1 First Order FPOs

Experimental unitary decomposition of single variable dependent quantum oper-

ations such as time evolution unitary of a constant Hamiltonian, controlled phase

gates (Cn-φ) and adiabatic evolution operators can be solved by performing first

order FPO. They yield a single line in the Fidelity profile. Here we investigate

how efficiently we can simulate these operations.

3.2.1.1 Controlled-Phase gate (C-φ)

Phase gate operation on target qubit, controlled by the state of control qubit is

an important operation in basic quantum information circuit. The unitary matrix

for this operation is given by (Eqn. 3.7),

C-φ =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 e−iφ

. (3.7)

The unitary operator decomposition by first order FPO is obtained as,

C-φ = R2(π, 5π/4) Uzz(φ/2) R2(π, (2π + φ)/4)

R1(π, π − φ/4) R1(π, 0).(3.8)

where Rn(θ1, θ2) and Uzz(φ) are given in Eqn. 3.2 and 3.4 respectively. The unitary

operator decomposition (Eqn. 3.8) has three φ dependent UOD parameters and

is exact (100% fidelity for all values of φ). We have performed experimental

verification of this in a two spin NMR system (Sec. 3.3).

3.2.1.2 Controlled-Controlled-Phase gate (CC-φ)

Controlled-Controlled-Phase gate applies a phase φ to the target when both control

qubits are in |1〉 state. The experimental unitary decomposition by first order FPO


is obtained as,

CC-φ = R1(π, 0) R2(π, φ/8) R3(π, φ/8) U13zz (φ/4)

R1(π/2, π − φ/4) U12zz (π/2) R1(π/2, 3π/2− φ/4)

U13zz (π − φ/4) R1(π/2, π/2) R3(π, π/2)U12

zz (π/2)

U23zz (2π − φ/4) R1(π/2, 0) R2(π, 0).

(3.9)

The decomposition (Eqn. 3.9) is exact (100% fidelity for all values of φ) has seven

φ dependent UOD parameters.

For C-φ and CC-φ gates, the φ dependence of the experimental unitary decom-

position are obtained as first order polynomials.

3.2.1.3 Quantum Simulation of three spin Heisenberg XY Hamiltonian

Heisenberg XY interaction (Eqn. 3.10) is a naturally occurring Hamiltonian in

various spin systems [69–71].

HXY = J(I1xI

2x + I1

yI2y + I2

xI3x + I2

yI3y ). (3.10)

Where J is the coupling strength. Spin product operators (Iα) and Pauli matrices

(σα) are related by Iα = σα/2 (where α = x, y, z). The time (t) evolution

operator of Heisenberg XY Hamiltonian is given by,

UXY = exp(−itHXY ) = exp(−iτ(I1xI

2x + I1

yI2y + I2

xI3x + I2

yI3y )). (3.11)

UXY is a single variable (τ = Jt) dependent three qubit unitary operator. The

UOD using first order FPO is obtained as (Eqn. 3.12),

UXY = U1 Uzz U2 Uzz U3 Uzz U2 Uzz U1,

U1 = R1(π/2, 0) R2(π/2, π/2) R3(π/2, 0),

U2 = R1(π/2, 0) R2(π/2, 0) R3(π/2, 0),

U3 = R1(θ1, θ2) R2(π, π/2) R3(π/2, 0),

Uzz = exp(−i π/2 (2I1z I

2z + 2I2

z I3z )),

(3.12)


0 4 8 1299.8

99.9

100

τ F

ide

lity

(%)

Figure 3.1: Fidelity profile of FPO (Eqn. 3.12) of Heisenberg XY interaction(Eqn. 3.11). Fidelities are calculated using Eqn. 3.3.

where,

θ1 = 5.171× 10−5τ 6 − 0.001392τ 5 + 0.01259τ 4

−0.03942τ 3 + 0.07208τ 2 − 0.5488τ + 3.152,

θ2 = 0.0002799τ 5 − 0.006301τ 4 + 0.04469τ 3

−0.09354τ 2 + 0.3353τ − 0.02239.

(3.13)

The fidelity profile (Fig. 3.1) for the decomposition Eqn. 3.12 has a least fidelity

> 99.95% for one complete period (τ=0 to 8.9) of UXY . While the Cn-φ FPOs

discussed in Sec. 3.2.1.1 and 3.2.1.2 are exact (100% fidelity), the FPO in the

present case is not exact, the fidelity here can be improved (if needed) by sub-

stituting more complex polynomial functions for θ1 and θ2. The decomposition

shown here is time symmetric – can be applied in the order left to right (increasing

time) or right to left – and uses less number of pulses (around 20 hard pulses) and

delays compared to the method by Zhang et al (around 40 hard pulses) [72].

3.2.1.4 Three Spin DM interaction

Dzyaloshinskii-Moriya (DM) interaction is an anisotropic antisymmetric exchange

interaction arising from spin-orbit coupling [73, 74]. Interaction of this type was

proposed by Dzyaloshinskii to explain the weak ferromagnetism of antiferromag-

netic crystals (α-Fe2O3, MnCO3)[73]. DM interaction is crucial in the description

of many antiferromagnetic systems [75–77].

Three spin DM interaction is given by (Eqn. 3.14) [74] ,

HDM =D

4(σ1

xσ2y − σ1

yσ2x + σ2

xσ3y − σ2

yσ3x), (3.14)


Figure 3.2: Pulse sequence for experimental simulation of three spin DMinteraction.

where D is the strength of the DM interaction. The time (t) evolution operator

of Dzyaloshinskii-Moriya Hamiltonian is given by,

UXY = exp(−itHDM) = exp(−iτ4

(σ1xσ

2y − σ1

yσ2x + σ2

xσ3y − σ2

yσ3x)). (3.15)

UDM is a single variable (τ = Jt) dependent three qubit unitary operator. The

UOD using first order FPO is shown in Fig. 3.2. The functional dependence of τ

on the rotation angles and phases (θ1 and θ2) of the pulse sequence (Fig. 3.2) is

obtained as,

θ1 = 2.527× 10−4τ 5 − 5.616× 10−3τ 4 + 0.03838τ 3

−0.068τ 2 + 0.2928τ + 1.568,

θ2 = −4.701× 10−5τ 6 + 1.253× 10−3τ 5 − 0.01101τ 4

+0.03076τ 3 − 0.04865τ 2 + 0.5207τ.

(3.16)

The decomposition shown here is time symmetric and uses only nearest neighbor

couplings.

3.2.1.5 Adiabatic Quantum Evolution

A typical example for first order FPO is the operators for Adiabatic evolution,

where the system is evolved under a slowly time varying Hamiltonian [32]. Adia-

batic Evolution Hamiltonian is given by,

H(S) = (1− S)Hi + SHf ,

where 0 6 S 6 1, Hi is the initial Hamiltonian and Hf is the final Hamiltonian.


Figure 3.3: Pulse Sequence decomposition of Eqn. 3.18.

0 0.25 0.5 0.75 199.99

99.995

100

S

Fid

elit

y (%

)

Figure 3.4: Fidelity profile of the Hamiltonian (Eqn. 3.18).

The unitary operator – U(S), Eqn. 3.17 – is a single variable unitary and hence

first order FPO can be used for solving adiabatic unitary decomposition.

U(S) = e−iH(S) (3.17)

Consider a case with Hi = (I1x + I2

x) and Hf = I1z I

2z , the adiabatic Hamiltonian

(HA) is,

HA = S(I1x + I2

x) + (1− S)I1z I

2z . (3.18)

The S dependent pulse sequence obtained using first order FPO is given in Fig.

3.3. The S dependent rotation angles and phases (θ1 to 4) are obtained as (Eqn.

3.19),

θ1 = 0.5904 S3 − 1.606 S2 + 1.544 S + 0.2609

θ2 = 0.1569 S3 − 0.1376 S2 + 0.7638 S + 0.002335

θ3 = 1.026 S3 − 2.839 S2 + 4.482 S + 0.4788

θ4 = −0.274 S3 + 0.1716 S2 + 0.1059 S − 0.003411

(3.19)

The decomposition is time symmetric and the fidelity is more than 99.99% for all

values of S.


3.2.2 Second order FPO

Second order FPO deals with two variable dependent quantum Unitary operations

and hence the fidelity profile is a plane. Here we explain this with an example of

simulation of Dzyaloshinskii-Moriya (DM) interaction (Sec. 3.2.1.4) in the pres-

ence of Heisenberg XY interaction, where the relative interaction strength (γ) and

time (τ) are the system parameters (Eqn. 3.20).

HDM = D(σ1xσ2y − σ1yσ2x), (3.20)

where D represents the strength of DM interaction. Here we present a generic

unitary operator decomposition to simulate the Hamiltonian – DM interaction in

the presence of Heisenberg XY interaction – in a two qubit system (Eqn. 3.21).

H(J,D) = J(σ1xσ2x + σ1yσ2y) +D(σ1xσ2y − σ1yσ2x), (3.21)

The unitary evolution operator of H(J,D) (Eqn. 3.21) is given by,

U(D, J, t) = exp(−iH(J,D)× t), (3.22)

Without losing generality, Eqn. 3.22 can be written as,

U(γ, τ) = exp(−i[(σ1xσ2x + σ1yσ2y)+

γ(σ1xσ2y − σ1yσ2x)] τ),(3.23)

where γ represents the relative ratio of interaction strengths (γ = D/J) and a

dimensionless quantity τ = J × t.As the first step, we performed FPO of U(γ, τ) with following assumptions – (a).

the range of τ is from 0 to 15 (this includes one complete period of 12.59; please

see Eqn. 3.27) , (b). the range of γ is from 0 to 1 and (c). The system Hamiltonian

is Hzz (Eqn. 3.1).

The optimized UOD (Eqn. 3.24) has seven SQRs and two system Hamiltonian

evolutions.

U(γ, τ) = R1(π2,−π

2) R1(π

2, θ2) R2(π, π) Uzz(

π4)

R1(θ1, θ2 + π2) R2(π − θ1, 0) Uzz(

π4)

R1(π2, θ2 + π) R2(π

2, π

2),

(3.24)


Figure 3.5: Fidelity profile of UOD given in Eqn. 3.24. (Fidelities are calcu-lated using Eqn. 3.3.)

where θ1 and θ2 (given in Eqn. 3.25) impart γ and τ dependence to UOD and Uzz

is given by Eqn. 3.4.

θ1 = [0.8423− 0.3455 Cos(1.117 γ)+

0.01806 Sin(1.117γ)] τ,

θ2 = 1.345 exp(−0.8731γ) + 1.796.

(3.25)

The fidelity profile of UOD is shown in Fig. 3.5. The minimum point in fidelity

profile is greater than 99.99 %. It should be noted that, the total length of UOD

(Eqn. 3.24) is invariant under γ or τ (with the assumption that all the SQRs are

instantaneous).

For generalizing the assumption on τ , we solved Eqn. 3.26 numerically and find

the period P(γ) of U(γ,τ) (Eqn. 3.27).

H(γ, τ + n× P (γ)) = H(γ, τ). (3.26)

where n is any positive integer.

P (γ) = 3.008 γ3 − 6.627 γ2 − 0.1498 γ + 12.59. (3.27)

Eqn. 3.27 has a maximum value of 12.59 at γ=0. Since the maximum value of

period is less than 15 (FOP performed till τ=15), UOD (Eqn. 3.24) can be used

for any value of τ . Same argument can be used for extending the range of τ to

−∞.


In order to incorporate the range of γ from 0 to ∞, we performed FPO for the

following operator (Eqn. 3.28),

U(Υ,∆) = exp(−i[Υ (σ1xσ2x + σ1yσ2y)+

(σ1xσ2y − σ1yσ2x)] ∆),(3.28)

where Υ and ∆ are dimensionless variables with range 0 to 1 (for Υ ) and 0 to 15

(for ∆). The optimized unitary decomposition using second order FPO is obtained

as,

U(Υ,∆) = R1(π2, π

2) R2(π

2, θ3) Uzz(

π4) R1(θ2 + θ3, 0)

R2(θ1, θ4) Uzz(π4) R1(π

2, π

2) R2(π

2, θ3),

(3.29)

where θ1 · · · θ4 (Eqn. 3.30) impart Υ and ∆ dependence to UOD.

θ = [0.09812 exp(−2.42Υ ) + 0.4023 exp(0.5524Υ )]∆,

θ1 = −θ + 3.142,

θ2 = θ − [1.242 exp(−0.9617Υ ) + 0.3546 exp(−0.1145Υ )],

θ3 = 1.259 exp(−0.957Υ ) + 3.479 exp(−0.0087Υ ),

θ4 = 1.256 exp(−0.959Υ ) + 1.912 exp(−0.0166Υ ).

(3.30)

Eqn. 3.28 satisfy the same periodicity relation as shown in Eqn. 3.27 and hence

can use the same reasoning for extending ∆ range from 0 to +∞.

For γ > 1, Eqn. 3.23 can be written as,

U(γ′, τ ′) = exp(−i[γ′(σ1xσ2x + σ1yσ2y)+

(σ1xσ2y − σ1yσ2x)] τ′),

(3.31)

where γ′ = 1/γ and τ ′ = γ × τ .

Eqn. 3.28 and Eqn. 3.31 are equivalent and hence UOD for Eqn. 3.23 can be

shown as,

U(γ, τ) =

Eqn. 3.23 if γ 6 1

Eqn. 3.28 if γ > 1(3.32)


Figure 3.6: 13C labeled Chloroform, two qubit NMR system, used for thequantum simulation of DM interaction (Eqn. 3.32) and controlled phase gate.13C and 1H act as qubits with zz interaction (Jzz=215.1 Hz) between them.The longitudinal and transverse relaxation time – T1, T2 (in seconds)– for 1Hand 13C are 21, 3.4 and 16, 0.29. The observed FID decay rates (T ∗2 ) are720 ms (1H) and 175 ms (13C).

Figure 3.7: 13C labeled CHFBr2 used for experimental verification of FPOof Heisenberg XY interaction (Eqn. 3.12). 1H , 13C and 19F forms a threequbit system with Hzz interaction. The longitudinal and transverse relaxationtime – T1, T2 (in seconds)– for 1H, 13C and 19F are 6.7, 1.4, 1.9, 0.71 and4.0, 0.7.

Combining all above steps forms most generic UOD of the Hamiltonian – DM

interaction in the presence of Heisenberg XY interaction.

3.3 Experimental

We have performed experimental verification of the FPOs – Controlled Phase gate,

simulation of Heisenberg XY interaction and DM interaction – in two and three

spin LS-NMR quantum information processor. Two spin LS-NMR system – 13C

labelled 13CHCl3 (dissolved in Acetone-D6) (Fig. 3.6) with 13C and 1H spins as

qubits (coupling interaction - Hzz; Eqn. 3.1) is used for simulating Controlled

phase gate (Eqn. 3.8) and DM interaction (Eqn. 3.32). Three spin LS-NMR

system – 13C labeled CHFBr2 (Fig. 3.7) is used for simulating Heisenberg XY

interaction (Eqn. 3.12). All experiments have been performed using Bruker AV-

500 NMR spectrometer at an ambient temperature of 300 K.


Experimental procedure for quantum computing in NMR is as follows,

(i) Initialization of quantum system – initial state creation is necessary for im-

plementing any quantum algorithm. In this chapter, we are initializing the

quantum system to computational basis state (pseudo pure state – [28–30])

and pure transverse magnetization. DM interaction simulation (Sec. 3.3.3)

used two spin (13C labelled CHCl3) pseudo pure state,

(ii) processing the initial state by evolving under different average Hamiltonian

[14] and

(iii) read-out by quantum state tomography [32].

There are several methods for creating pseudo pure states (PPS) in NMR from

equilibrium state [28–30]. We have utilized the spatial averaging technique [28] for

creating |0〉⊗n pseudo pure states as described in [78]. The pulse sequence for two

spin PPS creation is given in Fig. 3.8. This two spin PPS pulse sequence is used

for the experimental quantum simulation of DM interaction and entanglement

preservation (Sec. 3.3.3). A basic introduction to PPS and quantum state tomog-

raphy is explained in Introduction chapter (Chapter 1). Experimental verification

of controlled phase (Sec. 3.3.1) and simulation of Heisenberg XY interaction (Sec.

3.3.2) is performed on pure Y magnetization as the initial state. The quantum

state tomography is performed for the final state [78] read-out (Chapter 1).

External rf pulses of specific frequency, amplitude(A), phase and duration(t) are

used for implementing Rn(θ, φ) in NMR. Here A and t controls θ, rotation angle

of the pulse. zz unitary operator, Uzz(Θ) is implemented using time evolution (for

a time, t = Θ/πJ) of the spin states in NMR weakly coupling Hamiltonian (Eqn.

3.1) [14].

For comparing a series of experimental Ei=1...n and the theoretical Ti=1...nvalues, we have used ‘Average Experimental Deviation’ (AED), defined as (Eqn.

3.33),

AED =n∑i=1

|Ei − Ti||Ti|

. (3.33)


Figure 3.8: Pseudo pure state creation in a proton and carbon two spin sys-tem. Starting from thermal equilibrium, ρeq ∝ 4IHz + ICz , first (π/3)x pulseon proton covert 4IHz to 2(IHz −

√3IHy ). The first gradient (G1) kills the

transverse magnetization 2√

3 IHy and the remaining H magnetization, 2IHzis converted to

√2(IHz − IHy ) by a (π/4)x pulse. Evolution under J coupling

for a time 1/(2J) converts it to√

2(IHz − 2IHx ICz ), which again converted to

(IHz − IHx ) + (2IHx ICz + 2IHz I

Cz ) by (π/4)y pulse. The final gradient pulse (G2)

is applied to kill the transverse magnetization, yeilding the total magentizationas IHz + ICz + 2IHx I

Cz and is |00〉 pps. All other computational basis pps states

can be geneated by using π inversion pulses.

3.3.1 Controlled Phase (C-φ) gate

We have performed experimental verification of Controlled Phase (C-φ) FPO in13C labelled Chloroform CHCl3 (Fig. 3.6). The unitary operator decompositions

for Controlled phase gate is given in Eqn. 3.8.

Starting from initial state, ρin = I1y , controlled phase C-φ gate will modulate the

‘y’ magnetization as shown in Eqn. 3.34.

⟨I1y

⟩= Trace(C-φ× ρin × C-φ†) =

1

2(1 + Cos(φ)). (3.34)

We have experimentally measured this modulation and found it to match well

with theoretical result. Experimental and simulated plots with normalized⟨I1y

⟩intensity are shown in Fig. 3.9. We have observed an AED of 1.2 %, which

confirms the FPO.

3.3.2 Quantum Simulation of three spin Heisenberg XY

Hamiltonian

Experimental verification of the first order XY Hamiltonian FPO (Eqn. 3.12) is

performed in 13C labeled CHFBr2 three spin NMR system (Fig. 3.7). The spins


Figure 3.9: Experimental implementation of Controlled Phase (φ) gate forvarious phase angles (decompositions in Eqn. 3.8). ‘y’ magnetization –

⟨I1y

⟩(measured by partial tomograph[32] of the final state) – shows a φ modulationas shown. Experiments are performed in 13C labelled Chloroform (CHCl3; Fig.3.6) dissolved in CDCl3.

Figure 3.10: Experimental quantum simulation of Heisenberg XY Hamilto-nian (decompositions in Eqn. 3.12). Experiments are performed in 13C labeledCHFBr2 (Fig. 3.7). Second spin ‘y’ magnetization –

⟨I2y

⟩(measured by par-

tial tomograph[32] of the final state) – shows a periodic modulation with τ asshown.

1H, 13C and 19F forms a linear spin chain with NN coupling values JH,Czz = 224.5

Hz and JC,Fzz = −310.5 Hz (Fig. 3.7), if the two bond JH,Fzz = 49.7 Hz is refocused

during the entire experimental time.

Experimental verification of FPO is performed by using following scheme,

I1y + I2

y + I3y (initial state)

UXY (τ)−−−−→ Measure

τ modulated ‘y’ magnetization⟨I2y

⟩.

Second spin ‘y’ magnetization –⟨I2y

⟩– is modulated by τ as shown in Fig. 3.10.

Experimental results matches the theoretically simulated curve with an AED

(Eqn. 3.33) of 2.2%.


3.3.3 Quantum simulation of DM Hamiltonian and entan-

glement preservation

Here we study the entanglement dynamics (quantified by concurrence, Eqn. 3.36)

of a Bell state (Eqn. 3.35), evolving in DM + XY Hamiltonian (Eqn. 3.21).

|φ〉− =1√2

(|01〉 − |10〉). (3.35)

For a two qubit system (of qubits A and B), with density matrix ρAB, concurrence

(CAB) is defined as [79],

CAB = maxλ1 − λ2 − λ3 − λ4, 0, (3.36)

where λ1....4 are eigenvalue of the matrix ρABρAB with ρAB = (σy⊗σy)ρ∗AB(σy⊗σy).

Using the unitary operator decompositions shown in Eqn. 3.32, we have simulated

the Hamiltonian H(γ, τ) for γ=0.33, 0.66, 0.99. The experiments are performed

with 13C labelled Chloroform (Fig. 3.6). We have utilized spatial averaging tech-

nique [28] for creating |00〉 pseudo pure states [78]. Singlet state is created from

|00〉 state by using a Hadamard gate and controlled NOT gate (singlet state be-

longs to NOON state class and its creation is explained in Chapter 4).

Starting from singlet state (Eqn. 3.35), we have simulated DM + XY Hamiltonian

using the decomposition given in Eqn. 3.32. The τ modulated concurrence is

oscillating and is shown in Fig. Fig. 3.12(a). The experimental results (Fig.

3.12(a)) show a good agreement with the theoretical simulations. It is observed

that the value of concurrence oscillates for all values of γ and stays high for high

γ values.

Average experimental deviation (Eqn. 3.33) is observed to be 4.5%, 3.9 % and

3.1% for γ=0.33, 0.66, and 0.99 respectively. This experimental deviation can be

due to decoherence, static and rf inhomogeneities. Total experimental time was

9 ms, which is much shorter than the shortest T2 (290 ms) as well as T ∗2 (175 ms).

This rules out decoherence and static field inhomogeneity as the sources of error.


Figure 3.11: Experimental scheme used for entanglement preservation. τ isvaried from 0 to 3 in sixteen steps.

Entanglement Preservation

Entanglement plays a crucial role in several applications of quantum computation

and quantum information theory with best known applications in superdense cod-

ing [80] and quantum teleportation[81]. Preservation of entanglement is a crucial

element in the realization of quantum information processing. Hou et al. [82]

demonstrated a mechanism for entanglement preservation of a quantum state in

a Hamiltonian of the type given in Eqn. 3.22. Preservation of initial entangle-

ment of a quantum state is performed by free evolution interrupted with a certain

operator O, which makes the state to go back to its initial state. The operator

sequence for preservation is given in Eqn. 3.37.

O U(γ, τ) O U(γ, τ) ≡ I, (3.37)

where O = I1 ⊗ σ2z.

We have validated this entanglement preservation procedure using the experimen-

tal scheme as shown in Fig. 3.11. Starting from singlet state (|01〉−|10〉), we have

applied the preservation procedure as shown in Eqn. 3.37. Experiments have been

performed for three different values of relative interaction strength (γ), which are

γ=0.33, 0.66, 0.99. Varying τ (Fig. 3.11) from 0 to 3 in sixteen steps. The ex-

perimental results (Fig. 3.12(b)) shows excellent entanglement preservation. The

average experimental deviation of concurrence is less than 3%.

Conclusion

We have performed Fidelity Profile Optimization by Genetic algorithm and solved

single parameter and two parameters dependent quantum gates with experimental

verification in NMR. Exact φ dependent experimental unitary decompositions of

C-φ and CC-φ are solved using first order FPO. Unitary operator decomposition


(a)

(b)

Figure 3.12: (a).Entanglement (concurrence – Eqn. 3.36) dynamics of 13C−Hsystem under the Hamiltonian Eqn. 3.21, (b). Entanglement preservation ex-periment using Eqn. 3.37. Starting from singlet state, the concurrence sustainsat unity with the preservation procedure. (Symbols are same as in 3.12(a)).The inlet shows a blowup of the concurrence on a scale of 0.96 to 1 and τ from0 to 3. The entanglement varies between 0.97 and 0.995 for all the three valuesof γ.

for experimental quantum simulation of DM interaction in the presence of Heisen-

berg XY interaction is shown with second order FPO (for any relative strengths

of interactions γ and evolution time τ). Experimental gate time for this decom-

position is invariant under γ or τ , which can be used for relaxation independent

studies of the system dynamics. Using these decompositions, we have experimen-

tally verified the entanglement preservation mechanism suggested by Hou et al.

Entanglement being the most important resource, its preservation is crucial in

quantum information processing.

Chapter 4

Efficient Creation of NOON

States in NMR

4.1 Introduction

NOON state or Schrodinger cat state is a maximally entangled N qubit state with

superposition of all individual qubits being at |0〉 and being at |1〉(Eqn. 4.1),

|ψNOON〉 =|N|0〉, 0|1〉〉+ e−iφ |0|0〉, N|1〉〉√

2. (4.1)

NOON states [83, 84] – introduced by Barry Sanders [85] in 1989 – have received

much attention recently for its high precession phase measurements, which enables

the design of high sensitivity sensors in optical interferometry [86] and NMR [87].

Ability to make high precesion phase measurements of NOON state is very impor-

tant in quantum metrology and is exploited recently in NMR by Jones et al. [87].

Here we show an efficient and practical method for entangling and disentangling

spin-NOON states in NMR, which is,

(i) a minimal unitary operator decomposition in terms of experimental unitaries

– radio frequency pulses (single qubit rotation (SQR)) and delays (J evolu-

tion, basic two spin operation in NMR) – and has

(ii) good experimental fidelity by Genetic algorithm (GA) optimized composite

unitary operations (composite pulses and composite evolutions).

57

Chapter 4. Efficient creation of NOON states in NMR 58

This high fidelity decomposition effectively enables NOON state creation (NSC)

with minimal knowledge about the system parameters (allowed variation of rf

control field strength and coupling strength upto ±50%). We have addressed this

Efficient NOON state creation problem in spin chain with nearest neighbor (NN)

couplings and spin star topology (SST).

Even though our study is based on basic NMR weak coupling approximation

Hamiltonian (Eqn. 4.2), it can be extended to any system Hamiltonian (Hsys)

by ‘term isolation’ procedure by Bremner et al. [67, 68].

Hsys = 2πJIzIz (4.2)

Section 4.2 discuss the entangling and disentangling operator decomposition fol-

lowed by experimental implementation in three spin NMR quantum information

processor (Sec. 4.3).

4.2 Theory

4.2.1 NOON State Creation (NSC) operators

Starting from |0〉⊗N state, NOON state can be created by a Hadamard gate, UH

(Eqn. 4.3) and (N − 1) Controlled-NOT (UCNOT ; Eqn. 4.4) gates as shown in

Fig. 4.1[88].

UH =1√2

[1 1

1 −1

](4.3)

UCNOT =

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

(4.4)

The experimental unitary decomposition of UH and U i,jCNOT in a weakly coupled

Liquid State NMR (LS-NMR) system are given by,

UH = R(π/2, π/2)×R(π, 0), (4.5)


Figure 4.1: Quantum circuit for NOON state creation. N qubit system re-quires (N − 1) Controlled-NOT gates and a Hadamard gate.

and

U i,jCNOT = Rj(π/2, 0)×Rj(π/2, π/2)× U i,j

zz (π/2)

×Rj(π/2,−π/2)×Ri(π/2, π/2)

×Ri(π/2, 0)×Ri(π/2,−π/2),

(4.6)

where,

Rn(θ, φ) = exp(−iθ × [Cosφ Inx + Sinφ Iny ]), (4.7)

is single qubit rotation (SQR) operation on nth spin and,

Uzz(Θ) = exp(−i Θ 2IzIz). (4.8)

External rf pulses of specific frequency, amplitude(A), phase and duration(t) are

used for implementing Rn(θ, φ) in NMR, where A and t controls θ, rotation angle

of the pulse. Uzz(Θ) is implemented using, time evolution (τ = Θ/πJ) of the

NMR weakly coupling Hamiltonian (Eqn. 4.2).

The general NOON state circuit (Fig. 4.1) in a N spin NMR system with weak

coupling Hamiltonian requires,

Nr = 2 + 6× (N − 1), (4.9)

number of SQR rf pulses which cause experimental errors due to rf inhomogeneity

and imperfect calibration [89].

Using GA optimization, a minimal unitary operator decomposition for NSC is per-

formed for small spin systems (N < 10) by state to state optimization (explained

in Chapter 2) with |0〉⊗N as initial and NOON state as final state. Using these

decompositions we have identified the pattern of the SQRs and evolutions in NSC

and generalized to N spin cases – spin chain with NN couplings (Fig. 4.3(a))

and spin star topology (Fig. 4.3(b)). Most general spin system – spin cloud (Fig.


4.3(c)) – with random coupling between any random qubit pairs can be solved

as a spin chain model with NN coupling using interaction selection procedure by

Leung et al [90].

4.2.1.1 Spin chain with nearest neighbor couplings

Spin chain with nearest neighbor couplings (Fig. 4.3(a)) is one of the most studied

qubit network in quantum information processing [91, 92]. We studied the spin

chain NSC in following two cases,

(i) Spin chain with NN equal couplings (J i,i+1 = constant): The experimental

unitary operator decomposition in this case is obtained as,

Uchain = R(π/2, φ)×N−1∏n=1

[Un,n+1zz (π/2)R(π/2, φ)], (4.10)

All the (N + 1) pulses here are same (R(π/2, φ)) and global to all the spins

in the chain. This is the only case where NSC can be performed with global

external controls.

(ii) Spin chain with NN non-equal couplings : The unitary operator decomposi-

tions in this case is obtained as,

U ′chain =N∏n=2

[Rn(π/2, 0)× Un−1,nzz (π/2)]×

N∏m=2

[Rm(π/2, π/2)]×R1(θ, φ).

(4.11)

The decomposition has one global and (N − 1) local control pulses. This

corresponds to three pulses for Bell state (|00〉 + |11〉) creation (Eqn. 4.12,

Fig. 4.2),

U2002 = R2(π/2, 0)× U1,2zz (π/2)×R2(π/2, π/2)×R1(θ, φ), (4.12)

and five pulses for GHZ state (|000〉 + |111〉) creation in a heteronuclear

NMR system. Only one NN spin pair coupling will evolve during each delay

and is performed by interaction selection procedure by Leung et al. [90].


Figure 4.2: Pulse sequence for for two spin NOON state creation.

(a) (b) (c)

Figure 4.3: (a). Spin ising chain with nearest neighbor couplings. J i,i+1 isthe coupling between ith and (i+ 1)th spin, (b). Spin Star network. All B spinsare coupled with spin A with equal strength and there is no coupling betweenB spins, (c). A general multi-spin system (spin cloud).

4.2.1.2 Spin star topology

Spin star topology (Fig. 4.3(b)) has a central spin A connected to (N − 1) num-

ber of B spins with equal coupling and there is no coupling between any two B

spins. This is the case with more practical importance because of the availabil-

ity of experimental systems with sufficient size (N) such as trimethylphosphite

(N = 1 (31P ) + 9 (1H) = 10) and tetramethylsilane (TMS) (N = 1 (29Si) +

12 (1H) = 13). Experimental implementation of magnetic field sensing is per-

formed in trimethylphosphite by Jones et al [87, 93].

Using GA, the optimized unitary operator decomposition for spin star topology is

obtained as,

Ustar = RB(π/2, 0)× UA,Bzz (π/2)×RB(π/2, π/2)×RA(θ, φ). (4.13)

The system requires two global control pulses on spin-B and a single pulse on

spin-A for NSC.


4.2.2 Disentangling NOON states

Operators for disentangling NOON states are evaluated using the fact that product

of entangling operator (UE) and disentangling operator (UD) is identity (Eqn.

4.14).

UE × UD = I ⇒ UD = (UE)−1, (4.14)

where UE NSC operators.

Unitary decompositions for NSC operators (Eqn. 4.10, 4.11 and 4.13) combined

with inverse unitaries of SQRs (Eqn. 4.15) and the evolutions (Eqn. 4.16) can

be used for evaluating experimental unitary decomposition (UD) for disentangling

NOON states.

Rn(θ, φ)−1 = Rn(θ, φ+ π). (4.15)

U i,jzz (π/2)−1 = U i,j

zz (π/2)×Ri or j(π, 0). (4.16)

4.2.3 Robust NOON state creation

For an N qubit system, a complete knowledge about system parameters – control

field strengths (Fi), coupling strengths (Jj) – are necessary for the high fidelity

quantum information processing. Possible errors – uncalibrated control fields or

error in coupling value evaluations – and incomplete knowledge on system param-

eters can be avoided by finding unitary operator decompositions which has a self

compensatory mechanism for these errors.

The basic unitaries in an NMR pulse sequence are pulses (or SQRs) and delays

(or J evolutions). Robust unitary operator decomposition of NSC operators (UE

or UD) in terms of these SQRs and J evolutions are performed by replacing the

SQRs with composite SQRs and evolutions with composite evolutions. Composite

SQR (or composite pulse) [94, 95] is a sequence of adjacent pulses with optimal

amplitudes and phases whereas composite evolution operator [96] is a sequence

of adjacent evolution operators (here delays) interrupted by pulses. The opti-

mal delay values, amplitudes and phases for these composite unitaries (Ucom) are

evaluated using Genetic algorithm.


Composite unitary (Ucom) optimization using Genetic algorithm is performed by

maximizing (to unity) the fitness function,

F = Trace(Ucom(θp;φp;Dd)× Utar), (4.17)

where p varies from 1 toNp (total number of pulses) and d varies from 1 toNd (total

number of delays). Utar is the target unitary operator (SQR or evolution operator).

Now the optimization problem is to find the optimal values for 2×Np +Nd pulse

sequence parameters, which will maximize the function F. GA optimization of π/2

composite unitary (five pulse – requires optimal values for 10 parameters) took

around 15 minutes in a 3 GHz Intel processor.

Composite five pulse sequence for 90 (Eqn. 4.18) and 180 (Eqn. 4.19) SQR with

phase φ are obtained as (in the format Rotation Angle(Phase)),

90(φ) ≡ 91.60(45.00 + φ) : 153.67(30.00 + φ) : 177.20(235.81 + φ) :

183.19(342.47 + φ) : 128.72(120.00 + φ).(4.18)

and

180(φ) ≡ 178.81(102.57 + φ) : 171.90(334.93 + φ) : 186.35(291.44

+φ) : 175.12(76.76 + φ) : 186.70(17.39 + φ).(4.19)

The response of Eqn. 4.18 and Eqn. 4.19 against pulse angle error is shown in Fig.

4.4(a) and Fig. 4.4(b) respectively. Near 100% fidelity is observed for an error (in

pulse angle) upto ± 50 %.

The composite evolution (U czz(90)) is obtained as,

U czz(π/2) = P1(64.28)× Uzz(169)× P2(244.26)× Uzz(359.26)×

P2(64.26)× Uzz(179.4)× P1(244.29)(4.20)

P1(δ) = 108.99(298.79 + δ) : 172.05(77.49 + δ) : 108.99(298.79 + δ),

P2(δ) = 317.22(319.06 + δ) : 174.45(105.64 + δ) : 175.19(9.85 + δ) :

153.89(261.95 + δ).(4.21)


−100 −50 0 50 1000

50

100

Error in Flip Angle (%)

Fid

elity

(%

)

(a)

−100 −50 0 50 1000

50

100

Error in Flip Angle (%)

Fid

elity

(%

)

(b)

Figure 4.4: (a). 90 composite pulse (five pulse) response for flip angle error,(b). 180 composite pulse (five pulse) response for flip angle error.

0

80

8080

8090

9090

90 95

95

95

95

99

9999

99

99

90

Error in J (%)

Err

or in

flip

angl

e (%

)

−100 −50 0 50 100−100

−50

0

50

100

Figure 4.5: Flip angle and coupling error response of composite evolution(Eqn. 4.20).

P1 and P2 (Eqn.4.21) are in same format as shown in Eqn.4.18.

The response of composite evolution (Eqn. 4.20) is shown in Fig. 4.5 and has

robust (> 99%) response against coupling value and flip angle errors upto ±50%.

This gives an added advantage such that, complete knowledge on exact coupling

values or control field (Hrf ) strengths are not required for NSC. This advantage is


Figure 4.6: 13C labeled CHFBr2 used for experimental NSC. The spins 1H,13C and 19F forms a linear spin chain with NN coupling values JH,Czz = 224.5Hz and JC,Fzz = −310.5 Hz, if the two bond JH,Fzz = 49.7 Hz is refocused duringthe entire experimental time. The longitudinal and transverse relaxation time– T1, T2 (in seconds)– for 1H, 13C and 19F are 6.7, 1.4, 1.9, 0.71 and4.0, 0.7.

exploited in our experimental demonstration of three spin NOON state creation.

This idea can be used for cases like perfect state transfer [97], where engineered

spin chains are required.

4.3 Experimental

Experiments have been carried out in the three spin NMR system 13CHFBr2 (13C

labeled) (Fig. 4.6). The spins 1H , 13C and 19F forms a linear spin chain with NN

coupling values JH,Czz = 224.5 Hz and JC,Fzz = −310.5 Hz. The coupling between1H and 19F (JH,Fzz = 49.7 Hz) can be refocused during JH,F evolution [14]. All

experiments are performed in Bruker AV-500 NMR spectrometer at an ambient

temperature of 300 K.

Quantum information processing experiments in NMR starts with, (a). prepara-

tion of pseudo pure states [28–30], (b). processing the state by evolving under

different average Hamiltonians [14] and (c). read-out by quantum state tomogra-

phy [32]. We have utilized the spatial averaging technique [28] for creating |0〉⊗n

pseudo pure states as described in [78]. Pulse sequence for three spin PPS (|000〉)is shown in Fig. 4.7.

We have demonstrated NOON state creation using the operator decomposition for

spin chain (Eqn. 4.11) and spin star topology (Eqn. 4.13) in combination with

composite evolution sequence shown in Eqn. 4.20. Composite evolution (Eqn.

4.20) doesn’t require any prior knowledge on exact system variables (coupling


( . , )

, ,

,

,

,

,

Figure 4.7: Pulse sequece for three spin pesudo pure state creation.

(a) (b)

Figure 4.8: Quantum state tomography of three spin NOON state createdin (a). spin chain (Eqn. 4.11) and (b). spin star topology (Eqn. 4.13) incombination with composite evolution sequence shown in Eqn. 4.20. The labelsin XY plane 1,2...8 correponds to the computational basis states |000〉, |001〉,...... |111〉. The observed experimental fidelity is 95.9 % for spin chain and96.4 % for spin star topology.

values and control field strength). For the 1H −13 C −19 F linear spin chain with

NN coupling values JH,C = 224.5Hz and JC,F = 310.5Hz (sign flipped with an

extra π pulse), we have used the composite evolution tuned for 267 Hz, which can

give near 100% evolution response fidelity over a coupling range of 133 Hz to 401

Hz. For spin star topology, we have used 13C as the A spin and 1H and 19F as B

spins. The 1H −19 F coupling is refocused using a spin echo sequence [98].

The quantum state tomography is performed for the final state [78] and is shown

in Fig. 4.8(a) and 4.8(b). The observed experimental fidelity is 95.9% and 96.4%.

The remaining error is likely due to rf inhomogeneity. The contribution of deco-

herence to loss of fidelity is likely to be negligible, since the total experimental

time (≈ 30ms) is much less than the smallest T2 (= 700ms). A slightly lighter


fidelity in star topology is likely to be due to smaller number of pulses compared

to spin chain.

Conclusion

An efficient method for NOON state creation in NMR is demonstrated. The

unitary operator decomposition shown here is experimental friendly with a self

compensation mechanism on common experimental errors. Over a 50 % variation

in coupling value or spin control field strength is allowed, which effectively rules

out any strict constraints on system parameters of the spin system.

Chapter 5

Composite Operator Design

Using Genetic Algorithm

5.1 Introduction

A composite pulse is a cluster of nearly connected rf pulses which emulate the

effect of a simple spin operator with robust response over common experimental

imperfections [95]. Composite pulses are mainly used for improving broadband

decoupling [99–101], population inversion [102, 103], coherence transfer [104] and

in nuclear overhauser effect [105] experiments. Composite operator is a general-

ized idea where a basic operator (such as rotation or evolution of zz coupling)

is made robust against common experimental errors (such as miscalibration of rf

power or error in evaluation of zz coupling strength) by using a sequence of ba-

sic operators available for the system. Here using Genetic Algorithm, a global

optimization method based on the natural genetic evolution, we have designed

following composite operators,

(i) broadband rotation pulses,

(ii) rf inhomogeneity compensated rotation pulses and

(iii) zz evolution operator with robust response over a range of zz coupling

strengths.

We also performed rf inhomogeneity compensated Controlled NOT gate.

69

Chapter 5. Composite Operator design using Genetic Algorithm 70

5.2 Theory and Experimental

Single spin NMR Hamiltonian in presence of an external rf field is given by,

H = B1(Cos(φ)Ix + Sin(φ)Iy) + δIz, (5.1)

where B1 → strength of external rf in Hz, φ → Phase and δ → offset. The

evolution operator for this Hamiltonian (Eqn. 5.1) for a time τ is given by,

U(θ, φ,Ω) = exp(−iHτ),

= exp(−i(θ(Cos(φ)Ix + Sin(φ)Iy) + ΩIz)),(5.2)

where θ = B1τ and Ω = δτ .

We have solved U(θ, φ,Ω) (Eqn. 5.2) using Mathematica[106] yielding,

U(θ, φ,Ω) =

[Cos Θ− iΛΩ −iθΛe−iφ

−iθΛeiφ Cos Θ + iΛΩ

](5.3)

where Θ→√θ2 + Ω2

2and Λ→ Sin(Θ)

2Θ.

From the basic property of this operator matrix Eqn. 5.3, can be written as,

U(θ, φ,Ω) =

[α β

−β∗ α∗

](5.4)

where ‘*’ represents complex conjugate and α, β = Cos Θ − iΛΩ,−iθΛe−iφ.Hence α, β array is enough to represent matrix elements of a single spin unitary

operator.

U(θ, φ,Ω) ≡ α, β. (5.5)

The matrix product in this new representation can be shown as,

α1, β1 × α2, β2 = α1α2 − β1β∗2 , α1β2 + β1α

∗2 (5.6)

This reduction in computation of effective single spin operator can be used for

enhancing the optimization procedure. A Matlab function is written – Compos-

itePulseOperator (Code. 1) – for evaluating effective operation for N single spin

operations. Inputs of the function are arrays of N composite single spin operation

parameters – ‘Amplitude’, ‘Phase’ and ‘Offset’. Computational time advantage


1 f unc t i on E f f e c t i v eOpera to r=CompositePulseOperator (A1 , P1 , cs )E1=0.5*(A1.ˆ2 + cs . ˆ 2 ) . ˆ 0 . 5 ; E2=s i n (E1) . / ( 2*E1) ;

3 M=[( cos (E1)−1 i .* cs .*E2) (−1 i *exp(−1 i *P1) .*E2 .*A1) ] ;L=length (M) /2 ;

5 pdt=[M(1) M(L+1) ] ;f o r k=2:L ;

7 pdt=[pdt (1 ) *M( k )−pdt (2 ) * conj (M(L+k ) ) pdt (1 ) *M(L+k )+pdt (2 ) *conj (M( k ) ) ] ;

end9 Ef f e c t i v eOpera to r=pdt ;

CODE 1: Matlab Program for finding the effective operation of N compositeSQRs.

0 50 100 150 2000

20

40

60

Number of SQRs/Pulses

Tim

e A

dvan

tage

(γ)

Figure 5.1: Computational time advantage (γ, Eqn. 5.7) efficiency of effectiveoperator evaluation using ‘CompositePulseOperator ’ and Matlab builtin func-tion ‘expm’ [1]. The response is evaluated for a set of randomnly created Npulse composite sequence. (Matlab 7.14, 32 bit Ubuntu 12.04 OS with core i53.6 GHz Intel processor)

study (γ, Eqn. 5.7) of CompositePulseOperator over Matlab’s built in function for

matrix exponential – expm – is performed and is shown in Fig. 5.1.

γ = τ1/τ2, (5.7)

where τ1 → time for evaluating expm [1] and τ2 → time for evaluating Compos-

itePulseOperator).

In optimization point of view composite operator optimization can be categorized

as state to state (SS) and operator optimizations.

(i) State to state (SS) Optimization: Optimizing composite operator pa-

rameters for high fidelity final state generation from a specified initial state.

We have performed the following SS optimisations


(a) Excitation – SS optimization with thermal equilibrium (I iz) state as

initial state and I ix or I iy as the final state.

(b) Inversion – SS optimization with I iz state as initial state and −I iz as

the final state.

(c) Polarization Transfer – SS optimization with initial state → I1y and

final state → 2I1z I

2y

(ii) Operator Optimization : Optimizing composite operator parameters for

high fidelity spin operator. We have performed operator optimization for,

(a) π and π/2 pulses,

(b) zz - evolution operator (Uzz),

Uzz = exp(−iπ2σzσz), (5.8)

where σz is z - Pauli matrix,

(c) Robust Controlled-NOT [3] gate (UCNOT ),

UCNOT =

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

. (5.9)

GA optimization details of the Composite Pulse generation are given in the fol-

lowing,

5.2.1 Representation Scheme

Encoding solutions of the problem to individuals in genetic evolution is performed

using representation scheme. Here in composite pulse generation, we have used a

double array of length twice that of number of pulses (n) in the composite pulse.

5.2.2 Fitness Function

A fitness function is a particular type of objective function that describes the

optimality of a solution or individual. Flow of genetic evolution process of the


population is mainly controlled by fitness function and is crucial for any GA op-

timization. The effective operator of a composite pulse with an ‘n’ on-resonance

(Ω = 0) rf pulses is given by,

UCP = u1(θ1, φ1)× u2(θ2, φ2)....un(θn, φn), (5.10)

where uk(θk, φk) is the kth (k = 1, 2....n) rf pulse operator. Composite pulse

operator with robust response against rf inhomogeneity are optimized with the

fitness function (Frf ),

Frf =1

m

m∑p=1

Trace(UpCPrf × U

′tar), (5.11)

where,

UpCPrf = u1(θ1× (1 + rp), φ1)×u2(θ2× (1 + rp), φ2)....un(θn× (1 + rp), φn), (5.12)

where rp varies from −r to r in m steps and r mainly depends on number of rf

pulses in the composite pulse.

Composite pulse with broadband operator is performed with the fitness function

(Fbb),

Fbb =1

m

m∑p=1

Trace(UpCPbb × U

′tar), (5.13)

where UpCPbb is given by,

UpCPbb = u1(θ1, φ1,Ωp)× u2(θ2, φ2,Ωp)....un(θn, φn,Ωp), (5.14)

where Ωp is varies from −Ω to +Ω in m steps.

Composite pulse state to state optimizations (excitation and inversion) with robust

rf inhomogeneity are optimized with the fitness function (Frf ),

Frf =1

m

m∑p=1

Trace(UpCPrf × ρin × (Up

CPrf )† × ρ′tar), (5.15)

where UpCPrf is given in Eqn. 5.12.


Broadband state to state composite pulses (excitation and inversion) are optimized

with the fitness function (Fbb),

Fbb =1

m

m∑p=1

Trace(UpCPbb × ρin × (Up

CPbb)† × ρ′tar), (5.16)

where UpCPbb is given in Eqn. 5.14.

We also performed composite zz or evolution optimization for robust response

over a range of zz coupling values

Optimized composite pulses – in the format θ(φ) – and its experimetal verifi-

cation results are given below. The experimental verifications are performed in

H2O (1% H2O in D2O) . All experiments are performed in Bruker AV-500 NMR

spectrometer at an ambient temperature of 300 K. The strength of rf power used

is 50 KHz.

5.2.3 Broadband excitation pulses

We have performed optimization for three pulse, five pulse and seven pulse broad-

band excitation composite pulses by state to state optimization and are,

Three Pulse →274.6 (270.01 ): 206.23(90.01): 19.01(269.91),

Five Pulse →78.05(244.7): 198.22(2.58): 120.33(136.72): 130.74(17.35):

227.85(140.07) and

Seven Pulse →88.7(275.28): 112.66(294.07): 129.02(54.34): 61.24(4.17):

144.85(315.02): 128.37(66.65): 40.4(259.96).

The Matlab simulation excitation profiles of these composite pulses are shown in

Fig. 5.2.

The experimental verifications are performed by finding the proton spectral peak

intensity variation with change in offset and are shown in Fig. 5.3.


−2 −1 0 1 20

50

100

Relative Resonance OffsetFi

delit

y(a) Three pulse excitation

−2 −1 0 1 20

50

100

Relative Resonance Offset

Fide

lity

(b) Five pulse excitation

−2 −1 0 1 20

50

100


Fide

lity

(c) Seven pulse excitation

Figure 5.2: Matlab simulation of three pulse, five pulse and seven pulse broad-band excitation sequences. (State to state optimization with Iz as initial stateand Ix as final state.)

5.2.4 Broadband Inversion pulses


band excitation composite pulses by state to state optimization and are,

Three Pulse → 288.8(0): 197.6(180): 123(0): 34.2(180),

Five Pulse →89.56(58.28): 118.57(324.64): 330.87(162.59): 118.57(324.64):

89.56(58.28) and


−80 −40 0 40 800

50

100

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(a) Three pulse

−80 −40 0 40 800

50

100

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(b) Five pulse

−80 −40 0 40 800

50

100

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(c) Seven pulse

Figure 5.3: Experimental proton spectral intensity variation with differentoffset values of three pulse, five pulse and seven pulse broadband excitationsequences.

Seven Pulse →228.39(80.43): 67.28(74.67): 143.09(254.21): 61.94(65.14):

147.44(249.08): 203.67(66.17): 94.54(241.04).

The Matlab simulation inversion profiles of these composite pulses are given in

Fig. 5.4.


intensity variation with change in offset value and are given in Fig. 5.5.


−2 −1 0 1 2−100

0

80


delit

y

(a) Three pulse

−2 −1 0 1 2−100

0

80


Fide

lity

(b) Five pulse

−2 −1 0 1 2−100

0

80


Fide

lity

(c) Seven pulse

Figure 5.4: Matlab simulation of three pulse, five pulse and seven pulse broad-band inversion sequences. (State to state optimization with Iz as initial stateand −Iz as final state.)

5.2.5 Broadband π/2 pulses


band π/2 composite pulses by operator optimization. Unlike excitation pulses (SS

optimization which maximizes the transverse magnetization components starting

from thermal equilibrium state), broadband π/2 pulses act as π/2 rotation oper-

ator and doesn’t require any initial state conditions. The optimized broadband

π/2 pulses are,

Three Pulse → 328.86(2.22): 260.98(181.3): 22.51(354.8),


−80 −40 0 40 80−100

−50

0

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(a) Three pulse

−80 −40 0 40 80−100

−50

0

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(b) Five pulse

−80 −40 0 40 80−100

−50

0

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(c) Seven pulse

Figure 5.5: Experimental proton spectral intensity variation with differentoffset values of three pulse, five pulse and seven pulse broadband inversionsequences.

Five Pulse →120.35(180): 334.61(360): 161.65(180): 260.24(360):

122.11(360) and

Seven Pulse →139.65(4.7): 314.71(183.14): 192.47(3.66): 110.79(182.52):

311.19(1.54): 149.91(180.83): 20.6(347.33).

The Matlab simulation excitation profiles of these composite pulses are given in

Fig. 5.6.


intensity variation with change in offset value and are given in Fig. 5.7.


−2 −1 0 1 2

0

50

100


delit

y(a) Three pulse

−2 −1 0 1 20

50

100


Fide

lity

(b) Five pulse

−2 −1 0 1 20

50

100


Fide

lity

(c) Seven pulse

Figure 5.6: Matlab simulation of three pulse, five pulse and seven pulse broad-band π/2 sequences.

5.2.6 Broadband π pulses


band π composite pulses by operator optimization. Unlike inversion pulses (SS

optimization which maximizes the −Iz magnetization starting from thermal equi-

librium state), broadband π pulses act as π rotation operator and doesn’t require

any initial state conditions. The GA optimized broadband π pulses are,

Three Pulse → 58.08(0.42): 238.18(180.09): 360(0.06) ,


−80 −40 0 40 800

50

100

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(a) Three pulse

−80 −40 0 40 800

50

100

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(b) Five pulse

−80 −40 0 40 800

50

100

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(c) Seven pulse

Figure 5.7: Experimental proton spectral intensity variation with differentoffset values of three pulse, five pulse and seven pulse broadband π/2 sequences.

Five Pulse →136.15(61.39): 298.76(356.71): 138.12(141.33): 298.83(356.83):

136.1(61.54) and

Seven Pulse → 231.54(138.67): 239.31(366.12): 140.63(138.87):

97.51(41.98): 142.07(137.25): 237.87(364.3): 231.31(134.41).

The Matlab simulation inversion profiles of these composite pulses are given in Fig.

5.8. The experimental verifications are performed by finding the proton spectral

peak intensity variation with change in offset value and are given in Fig. 5.9.


−2 −1 0 1 2−100

0


delit

y

(a) Three pulse

−2 −1 0 1 2−100

0


Fide

lity

(b) Five pulse

−2 −1 0 1 2−100

0

80


Fide

lity

(c) Seven pulse

Figure 5.8: Matlab simulation of three pulse, five pulse and seven pulse broad-band π sequences.

5.2.7 rf inhomogeneity compensated composite pulses

We have performed rf inhomogeneity compensated composite pulse optimization

for seven pulse excitation, nine pulse excitation, five pulse π/2 and five pulse π

composite sequences and are obtained as,

Seven pulse excitation →185.94(323.68): 349.1(287.64): 367.24(22.65):

289.92(344.5): 21.3(115.32): 637.28(153.15): 712(35.48),


−80 −40 0 40 80−100

−50

0

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(a) Three pulse

−80 −40 0 40 80−100

−50

0

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(b) Five pulse

−80 −40 0 40 80−100

−50

0

Offset ∆ω (KHz)

Nor

mal

ised

Inte

nsity

(c) Seven pulse

Figure 5.9: Experimental proton spectral intensity variation with offset valuesof broadband three pulse, five pulse and seven pulse π composite pulses.

Nine pulse excitation →885.06(232.85): 228.48(300.87): 128.53(157.7):

301.86(97.7): 759.06(213.3): 884.46(42.83): 819.03(120.31): 702.6(45.25):

1077.55(217.84),

Five pulse π/2→228.61(105): 383.51(90): 442.24(295.81): 457.19(402.47):

321.25(180) and

Five pulse π→465.94(17.39): 437.05(76.76): 465.06(291.44): 429.01(334.93):

446.26(102.57).


−1 −0.5 0 0.5 10

50

100

Error in RFFi

delit

y(a) seven pulse

−1 −0.5 0 0.5 10

50

100

Error in RF

Fide

lity

(b) nine pulse

Figure 5.10: Matlab simulation of seven pulse and nine pulse composite exci-tation sequences with robust rf inhomogeneity profile.

The Matlab simulation of these composite pulses are given in Fig. 5.10 and Fig.

5.11.


intensity variation with change in rf and are given in Fig. 5.12.

5.2.8 Composite Polarization transfer and Evolution op-

erator

We have performed composite polarization transfer optimization (state to state

transfer with initial state I1y and final state 2I1

z I2x) as well as composite zz evolution

operator (Uzz, Eqn. 5.8). Unlike composite pulses discussed above, composite

delay operator optimization is performed over zz evolution operators (or delays)

and hard pulses for near 100% fidelity for a range of zz coupling strength.

The GA optimized composite operator sequence is given below (in the format

θ(φ) − d ), where d (in bold) represents the zz evolution delay in terms of

angle.


−1 −0.5 0 0.5 150

75

100

Error in RF

Fide

lity

(a) Five pulse π/2

−1 −0.5 0 0.5 10

50

100

Error in RF

Fide

lity

(b) Five pulse π

Figure 5.11: Matlab simulation of five pulse π/2 and π composite sequenceswith robust rf inhomogeneity profile.

−0.8 −0.4 0 0.4 0.8

50

100

Error in rf

Nor

mal

ised

Inte

nsity

(a) Five pulse π/2

−0.8 −0.4 0 0.4 0.8

50

100

Error in rf

Nor

mal

ised

Inte

nsity

(b) Seven pulse excitation

Figure 5.12: Experimental proton spectral intensity variation with rf valuesof five pulse π/2 and seven pulse excitation.


0 2 4 60

50

100

Error in JFi

delit

y(a) Polarization transfer (initial state: I1y and final state: 2I1z I

2x)

−1 −0.5 0 0.5 150

75

100

Error in J

Fide

lity

(%)

(b) Uzz (Eqn. 5.8) operator

Figure 5.13: Composite operator sequence for for J compensation.

The optimized operator sequence obtained for polarization transfer is,

Polarization Transfer →177.5 - 358.17(77.11) - 88.99 - 161.95(72)

- 177.52 - 296.4(57.73) - 177.52 - 73.44(348.41) - 177.51 - 97.67(32.24)

- 90(90).

The Matlab simulation of this polarization transfer profile with change in J cou-

pling strength is shown in Fig. 5.13(a). The composite polarization transfer

operator has a near 100% fidelity for error in J upto ±50% and shows a periodic

repetition as shown.

The optimized composite Uzz operator (Eqn. 5.8) is obtained as,

evolution operator →85.52(64.28) - 168.97 - 203.25(64.26) - 359.26

- 200.7(244.26) - 179.4 - 83.2(244.29) - 91.53 - 349.57(64.29).

The Matlab simulation of Uzz operator profile with change in J coupling strength

is shown in Fig. 5.13(b). The composite polarization transfer operator has a near

100% fidelity for error in J upto ±50%.


Figure 5.14: Pulse Sequence for Controlled NOT gate.

−1 −0.5 0 0.5 125

50

75

100

Error in RF

Fide

lity

(%)

Figure 5.15: Inhomogeneity response profiles of controlled NOT gate.

5.2.9 RF inhomogeneity compensated Controlled NOT gate

Controlled NOT (CNOT) gate is a basic two qubit gate. CNOT applies a NOT

operation on target qubit based on the state of the control qubit. Any quantum

circuit can be simulated to an arbitrary degree of precision using a combination of

CNOT gates and single qubit rotations [3]. The pulse sequence used in NMR to

perform CNOT gate is shown in Fig. 5.14. We have performed GA optimization

for a robust CNOT gate operation in a range of rf intensity by replacing all rf

pulse with composite rotation pulses.

The rf compensated Controlled NOT gate operator (UCNOT – Eqn. 5.9) is obtained

as,

Controlled-NOT gate →349.63(343.94)1: 349.63(318.43)1:

173.51(127.44)1: 352.33(106.18)1: 173.51(84.92)1: 349.63(343.94)2:

349.63(318.43)2: 173.51(127.44)2 : 352.33(106.18)2: 173.51(84.92)2

- 90 - 349.63(343.94)2: 349.63(318.43)2: 173.51(127.44)2:

352.33(106.18)2: 173.51(84.92)2.


The composite sequence has 15 hard pulses and single delay. The rf response of

composite pulse is shown in Fig. 5.15. The fidelity stays near 100% for rf deviation

upto ±50%.

5.3 Conclusion

Using Genetic Algorithm Optimization, we have generated composite operators

– composite pulses and composite evolution operators – which is robust against

common experimental errors. We experimentally verified all the composite pulses

in H2O sample. We also optimized controlled NOT gate, which is robust against

±50% error in control field strength.

The theory of robust composite pulses in NMR starts by the article “NMR Pop-

ulation Inversion Using a Composite Pulse” by Malcolm Levitt and Freeman[95]

(1979). Ever since, fine tuning of composite pulses are performed for different

applications in NMR using numerical and analytical methods [99–105, 107–112].

The composite pulse sequences for brodband and rf inhomogeneity compensation

shown in this chapter may not show a good improvement over existing sequences,

still GA optimization can be used for making robust complex operators as shown

in the case of polarization transfer operator and Controlled NOT gate.

Chapter 6

Fast and Accurate Quantification

Using GA Optimized 1H −13 C

Refocused Constant-Time INEPT

6.1 Introduction

NMR spectroscopy provides a powerful and convenient tool for quantitative analy-

sis of newly synthesized compounds and natural products [113–115]. NMR spectra

used for quantification must have (i) well resolved peaks with unambiguous spec-

tral assignment and (ii) non-modulated peak intensities. In general spectral peak

intensities are modulated by system parameters such as coupling strength, relax-

ation, mobility of molecule and number of protons attached. These quantification

criteria naturally select 1H decoupled 13C NMR for complex mixture quantifica-

tion. Large experimental time, because of the less sensitive 13C nuclei, is the major

limitation of traditional 13C quantification experiment. To overcome this, experi-

ments utilizing polarization transfer from abundant high sensitive protons to 13C

(APT, NOE, INEPT) [116–118] are used, but they are sensitive to system param-

eters. By averaging over a number (n) of experiments with different polarization

transfer response, experiments like INEPT or refocused INEPT (R-INEPT) can

be made less sensitive to coupling strength and the number of attached protons

[119, 120]. This forms an optimization problem, where uniformity of the transfer

response over J coupling and average polarization transfer decides the convergence

criterion (or quantification efficiency).

89

Chapter 6. GA Optimized Quantitative INEPT (GAQIC) 90

This problem was recently addressed by Makela et al. [121], who devised refocused

constant time INEPT (Q-INEPT-CT) experiment. Makela et al. optimized a

delay of eight scans, and obtained a nearly uniform polarization transfer for CH1,

CH2 and CH3 carbons over a coupling range of 115-170 Hz, allowing accurate

quantification of the relative concentration of various compounds in a mixture.

Makela et al. used quasi-Newton non-linear regression algorithm in Microsoft

Excel solver routine for optimization of the values of the delays in a delay set.

Since experimental time is the major concern, we have addressed the problem

with various optimization constraints, which can improve the average polarization

transfer and hence reduce the experimental time. We have used Genetic Algorithm

(GA) optimization (Matlab®) for finding the optimized delay sets. We name these

experiments as GA-(Q-INEPT-CT) or in short GAQIC.

There are several potential solutions for J compensation in literature [122–124].

Broad-Band INEPT (BB-INEPT) by Bodenhausen et al. [122] and 90/45 INEPT

by A.M Torres et al. [123] uses composite evolution (sequence of evolution delays

interrupted by pulses) for J compensation. In optimization point of view, both

GAQIC and BB-INEPT controls the pulse sequence parameters to obtain an opti-

mized transfer response. BB-INEPT controls the pulse sequence parameters from

single experiment for J compensation, whereas GAQIC perform this by control-

ling pulse sequence parameters from a set of experiments, which provides a bigger

search space for GAQIC.

For a given range of J coupling, the response of BB-INEPT is not uniform for

CH1, CH2 and CH3, which rules out the possibility of using all the carbon signals

for quantification (averaging over many peaks is always an advantage). The pulse

sequence length as well as number of pulses also limit BB-INEPT for quantifica-

tion. Spectral edited quantitative delay sets (S1 and S2) are optimal in terms of

number of pulses and length, hence it is advantageous over Nyberg et al. [125] for

quantification.

Sec. 6.2 describes Genetic Algorithm optimization of delay sets followed by ex-

perimental implementations in Sec. 6.3. We also performed quantitative spectral

editing technique based on average polarization transfer and discuss it in Sec.

6.2.4.


6.2 Theory

Proton-carbon polarization transfer functions for methine (CH1), methylene (CH2)

and methyl (CH3) groups in refocused INEPT experiment [14, 126] are given by,

PCH1 =Sin(πJCH∆T )× Sin(πJCH∆R),

PCH2 =Sin(πJCH∆T )× 2Sin(πJCH∆R)× Cos(πJCH∆R),

PCH3 =Sin(πJCH∆T )× 3Sin(πJCH∆R)× Cos2(πJCH∆R),

(6.1)

where ∆T and ∆R are polarization transfer and refocusing delays. By averaging

over a number of experiments, n, with optimal delay set ∆T , ∆R, J dependence

of the polarization transfer can be approximated to a constant function (Eqn. 6.2).

n∑m=1

Pmi (J115Hz−170Hz) ≈ Ki, a constant,

where i = CH1, CH2, CH3.(6.2)

Only those delay sets which satisfies the constant function approximation (Eqn.

6.2) (or having a uniform polarization transfer response) can be used for quantifi-

cation.

The GAQIC pulse sequence is shown in Fig. 6.1. The delays ∆1, ∆2 and ∆3 (in

pulse sequence) are related in such a way that the chemical shift evolution during

transfer and refocusing delays are refocused (Eqn. 6.3). Using Eqn. 6.3, it can

be shown that the effective J-coupling evolution during INEPT and refocusing

segments are 2∆T1 and 2∆R1 respectively.

∆i1 + ∆i2 = ∆i3, (6.3)

Where i = T,R. The delays ∆i1, ∆i2 and ∆i3 are given by,

∆i1 = ∆/2,

∆i2 = ∆max/2−∆/2,

∆i3 = ∆max/2.

(6.4)

Where ∆ is the delay value from optimized delay set and ∆max is the maximum

delay in the optimized delay set. These choices of delay values will make sure the


Figure 6.1: GAQIC pulse sequence. This pulse sequence is same as in Makelaet al. [121]. Narrow black and wide gray bars represents 90o and 180o pulsesrespectively. Multiple narrow white bars in the 13C channel is 180o compositepulse, which reduce the off-resonance effects as shown in [121]. The first two 90o

pulses on carbon channel (purge) are to remove the native 13C magnetization.Gradient strengths are 45.4 G/cm (for G1 and G2) and 53 G/cm (for G3)with 4ms (for G1 and G2) and 200 us (for G3). We have used WALTZ-16decoupling during acquisition [56]. Phase cycling is as follows: φ1 = y, -y; φ2 =x, x, -x, -x; φrec = x, -x, -x, x.

total experimental evolution time (Tevol) is constant (Eqn. 6.5).

Tevol = (∆T1 + ∆T2 + ∆T3) + (∆R1 + ∆R2 + ∆R3)

= ∆Tmax + ∆Rmax = Constant.(6.5)

Keeping Eqn. 6.2 as converging criterion for optimization, we used Genetic Algo-

rithm for finding the optimal delay sets for (n =) 8 and 16 scan experiments. The

range of J coupling values for delay set optimization is set to (115 Hz - 170 Hz)

(C-H direct bond coupling of most of the chemical structures falls in this range).

The first step in GA optimization is defining a proper representation scheme for

encoding the solution of the problem to individuals in genetic evolution. In our

representation scheme, we have selected a floating point array of length 2n (n-

number of scans) for representing an individual. We have used a population of

size of 100 individuals for 1000 generations. Initial population is created by a

random array generator with a restriction on max value (posed by maximum

experimental time limit). Individual selection for next generation is performed by

Roulette wheel selection method [127]. We have used uniform mutation with rate

0.05 and crossover function ‘intermediate’ with ratio 1. ‘Elite count’ is set to 2

and ‘crossover fraction’ to 0.8 [127]. In a 3.6 GHz Intel-i5 processor with Ubuntu

12.04 OS, optimization takes around 2 hrs to converge for 8 scan experiment.

A fitness function is a particular type of objective function that prescribes the

optimality of a solution or individual. The basic fitness function of GAQIC (or


‘quantification efficiency’ of a delay set) is a function of average min-max differ-

ence (MMD) (difference between minimum and maximum points in the polariza-

tion transfer response) and average polarization transfer (APT) (mean value of

the transfer response). Min-max difference is a direct measure of uniformity (∝accuracy of quantification) of the transfer response whereas average polarization

transfer is inversely proportional to total experimental time.

Derived from Eqn. 6.2, we have used three different fitness functions with various

constraint levels. (A) near uniform and equal polarization transfer response collec-

tively for all the three types of carbons (B) near uniform and separate polarization

transfer response individually for each carbon and (C) near uniform polarization

transfer response for CH2 and CH3 carbons taken together.

6.2.1 Near uniform and equal response for CH1, CH2 and

CH3 transfer functions

The GA optimization fitness function in this case searches for a n scan experiment

delay set with near uniform and equal polarization transfer response for CH1, CH2

and CH3. Eqn. 6.2 with,

KCH1 = KCH2 = KCH2 , (6.6)

forms the fitness function constraint of the optimization.

We have optimized case A - GAQIC delay sets for (n=)8 and (n=)16 scan ex-

periments. The experiments are named as A8 (eight scan), A8´ (eight scan time

restricted (5.5 ms)) and A16 (sixteen scan). The optimized delay sets (∆T , ∆R)(in ms) are obtained as,

A8 → (2.1955, 2.7088) (1.9759, 1.235) (1.9657, 1.2291) (2.0504, 1.2121)

(4.7903, 3.7006) (8.9549, 4.4513) (4.0376, 3.7339) (1.9959, 1.1984),

A8´ → (5.5, 3.5836) (1.5283, 1.7226) (1.4814, 1.8069) (1.5419, 1.7993)

(1.4037, 1.7661) (1.5982, 1.8304) (5.5, 3.5999) (1.4943, 1.839) and

A16 → (2.0019, 1.4796) (8.9623, 4.6409) (1.8833, 1.6614) (5.8929, 3.9368)

(2.3277, 1.6185) (1.5878, 1.7082) (1.916, 1.0237) (5.6642, 3.9257) (3.9001,


3.7054) (1.7713, 1.0008) (0.54152, 9) (1.7067, 2.2225) (1.8876, 2.4176) (1.9443,

1.6098) (3.8368, 3.7403) (1.9736, 1.6823).

Average polarization transfer obtained for case A is ≈ 45%.

6.2.2 Near uniform and separate response for CH1, CH2

and CH3 transfer functions

The delay set optimization in this case search for near uniform polarization transfer

responses separately for each CH1, CH2 and CH3 carbons. The constraints of the

fitness function is given by Eqn. 6.2 with,

KCH1 6= KCH2 6= KCH3 . (6.7)

Compared with case A optimization, above constraints forms a bigger search space

and hence improved delay sets (with better MMD and APT). The optimized delay

sets in this case are named as B8 (eight scan) and B8´ (eight scan time restricted

(6 ms)). The optimized delay sets (in ms) are obtained as,

B8 → (2.943, 1.1792) (8.9563, 2.7485) (4.6173, 2.0081) (2.9618, 1.1761)

(2.9845, 1.1799) (4.2151, 3.0045) (2.9557, 1.1698) (2.9543, 1.1797) and

B8´ → (2.4107, 1.266) (2.8095, 0.9906) (1.7982, 1.2918) (5.9998, 2.6305)

(1.707, 1.5899) (5.6002, 3.1036) (1.958, 1.3282) (3.8098, 1.4394).

Average polarization transfer for case B delay set is ≈ 60%. Group wise aver-

age polarization transfer for B1 and B2 are 41.37CH1 , 62.73CH2 , 76.89CH3 and

45.23CH1 , 63.38CH2 , 73.349CH3 respectively.

6.2.3 Near uniform and separate response for CH2 and CH3

only

The delay set optimization in this case is a special case of case B (Sec. 6.2.2)

optimization. Here we loosen up the constraint level (Eqn. 6.2 and Eqn. 6.8) to


(a) (b)

(c) (d)

Figure 6.2: Comparison of polarization transfer response of refocused INEPTand GAQIC experiments. (a) refocused INEPT with J=145 Hz, GAQIC ex-periments; (b) case A (delay set A8 ), (c) case B (delay set B8 ) and (d) case C(delay set C8 ). Case C optimization has the best quantification efficiency with‘min-max difference’ of < 1 %. Average polarization transfer (which effectivelydetermines the experimental time) for case A, case B and case C delay sets are≈ 45%, 60% and 70% respectively.

improve the search space and hence results in delay sets with better quantification

efficiency.

KCH2 6= KCH3 . (6.8)

There is no constraint on CH1 polarization transfer. Compared to case A and

case B delay sets, we have obtained case C delay set with (i) average polarization

transfer > 70%, (ii) max-min difference < 1% and (iii) maximum delay of 5.5 ms.

However case C delay set is not suggested for those cases with more number of

CH1 groups in the compound mixture. The optimized delay set (in ms) obtained

is,

C8 → (2.2947, 1.2272) (2.9719, 1.7409) (3.0456, 1.7166) (5.5, 2.9666)

(2.403, 1.1468) (5.5, 3.0401) (1.7251, 1.0497) (2.4688, 1.0348).


Method Delay Set MMD (%) APT (%)

GAQIC

A8 2.98 44.47A8´ 8.45 47.46A16 2.13 44.23B8 1.25 60.33B8´ 2.77 60.70C8 0.83 70.28

QINEPT-CTH8 3.64 44.65H8´ 12.73 46.83

R-INEPT 49.88 78.03

Table 6.1: Min-Max Difference (MMD) and Average Polarization Transfer(APT) of all optimized delay sets and R-INEPT.

Average polarization transfer for case C delay set is ≈ 70%. Group wise average

polarization transfer for C8 is 66.1CH2 , 74.46CH3.

A comparative study of polarization transfer responses of refocused INEPT, case

A, case B and case C are shown in Fig. 6.2(a), 6.2(b), 6.2(c) and 6.2(d) respectively.

Average polarization transfer, which effectively determines the experimental time

for case A, case B and case C delay sets are ≈ 45%, 60% and 70% respectively

(detailed quantification efficiency of all the optimization delay sets are shown in

Tab. 6.1). The non-optimized R-INEPT (Fig. 6.2(a)) has MMD of ≈ 50% with

APT 78%, whereas case C optimized delay sets has MMD < 1% with APT 70%.

We have also compared our optimized delay sets with ‘original modulation’ (H8 )

and ‘time restricted modulation’ (H8´) of Makela et al. (Tab. 6.1).

6.2.4 Spectral editing by delay sets

A uniform, near zero polarization transfer response (Eqn. 6.9) for any of the

chemical group (CH1 or CH2 or CH3), will effectively remove those peaks in

refocused-INEPT spectrum. In the quantitative case, we have used the following

constraint,

KCH2 = 0 & KCH1 = KCH3 , (6.9)

which simplifies the spectrum (by removing the signals from CH2 carbons) without

losing the quantitative information. Compared to DEPT, the polarization transfer

response here is averaged over a number of experiments and has a near uniform


(a) (b)

Figure 6.3: Polarization transfer responses for quantitative spectral editingdelay sets (a). four scan (S4 ) (3.1187, 4.988) (5.8421, 6) (2.9747, 1.3583)(2.24, 3.0151), (b). sixteen scan (S16 ) (1.2241, 3.2027) (2.5979, 1.4173)(5.9911, 6.8728) (9.7327, 8.9775) (1.7114, 2.4693) (6.1213, 7.6599) (2.6226,4.9325) (2.8015, 1.5952) (2.1121, 4.9178) (5.3706, 7.9835) (2.17, 2.9369) (1.0491,3.0062) (2.7459, 1.0893) (3.4169, 5.5314) (1.7431, 4.403) (1.7297, 5.4263) (inms).

behavior over a J coupling range 115 Hz to 170 Hz. Using the constraints given in

Eqn. 6.9, we have optimized two delay sets, which are S4 (four scan) (Fig. 6.3(a))

and S16 (sixteen scan) (Fig. 6.3(b)).

It may be mentioned that the optimized delay sets given here will also work for

basic refocused INEPT sequence [118, 126]. However, the sequence of Fig. 6.1 as

suggested by Makela et al. also takes care of offset and inhomogeneity effects and

is a constant time experiment.

Additionally we have generated angle sets (θ = Jτ) for quantification in a mixture

with only one type of chemical group (SI1 or SI2 or SI3). We have named these

experiments as SI1-GAQIC (Fig. 6.4(a)), SI2-GAQIC (Fig. 6.4(b)) and SI3-

GAQIC (Fig. 6.4(c)). Optimization is performed over a parameter set of eight scan

experiment (8×2 delays) and has average polarization transfer ≈ 80%. The delay

sets can be used for any types of spin system such as CHn or NHn (n = 1, 2, 3).

The width of uniform profile can be increased by compromising over the maximum

polarization transfer.


0 0.5 1 1.5 20

0.4

0.8

J/JSI

Res

pons

e

(a) SI1-GAQIC

0 0.5 1 1.5 20

0.4

0.8

J/JSI

Res

pons

e

(b) SI2-GAQIC

0 0.5 1 1.5 20

0.4

0.8

J/JSI

Res

pons

e

(c) SI3-GAQIC

Figure 6.4: Angle sets (θ = Jτ) for (a) SI1 (1.5340, 1.5325), (2.7269,2.7205), (3.3392, 3.3238), (1.5250, 1.5267), (1.5269, 1.5298), (1.5244, 1.5271),(1.5344, 1.5318), (1.5152, 1.5238), (b) SI2 (3.3346, 1.6642), (1.5229,0.7640), (1.5270, 0.7624), (2.7182, 1.3651), (1.5273, 0.7633), (1.5343, 0.7671),(1.5338, 0.7643), (1.5213, 0.7618) and (c) SI3 (1.3839, 0.4803), (2.4315,1.3210), (1.4751, 0.6842), (1.2293, 1.7333), (1.3692, 0.6453), (1.0413, 0.6375),(2.0486, 0.9171), (0.8760, 0.5637).

6.3 Experimental

Experimental implementation (Fig. 6.1) of GAQIC - case A, B and C delay

sets have been performed in a mixture of 4-Penten 2 ol (P) (Fig. 6.5(a)), α-

methoxy phenyl acetic acid (M) (Fig. 6.5(b)) and 1-2-dimethyl cyclohexane (D)

(Fig. 6.5(c)) dissolved in CDCl3. All the experiments are performed in Bruker

AV-500 spectrometer at 300 K. Proton decoupled 13C spectrum of the mixture

with peak assignments is shown in Fig. 6.6. The GAQIC pulse sequence (Fig.

6.1) is identical to the pulse sequence given by Makela et al. [121]. The composite

pulses effectively takes care of rf inhomogeneity effects and detailed analysis of

offset effects are contained in Fig. 3 of [121] and the same is applicable here. All

FIDs are acquired for 1 second and apodized using an exponential function (0.3

Hz line broadening). All the spectra were analyzed using custom written matlab

program. The relative quantities of compound-M (40 mg) and compound-D (20

µl) with respect to compound-P (10 µl) are obtained as,

RQ21 =moles of compound-M

moles of compound-P= 2.47,

RQ31 =moles of compound-D

moles of compound-P= 1.42.

(6.10)

Experimentally measured relative quantities (RQ21 and RQ31) using case A (three

experiments: A8, A8´, A16 ), case B (two experiments: B8, B8´) and case C (A8 )

optimized delay sets are shown in Tab. 6.2. Inverse gated 1H decoupled 13C


(a) (b) (c)

Figure 6.5: Samples used for GAQIC experiment (a) 4-Penten 2 ol (P), (b) α-methoxy phenyl acetic acid (M) and (c) 1-2-dimethyl cyclohexane (D).

Figure 6.6: Equilibrium spectrum of the compound mixture (Fig. 6.5). Theassignments of various peaks (derived from DEPT experiments) are indicatedon the top of each resonance.

experiment with 45 excitation pulse is used as traditional 13C quantitative (Q)

measurement [121]. Average over all signals was taken since it will reduce the

random errors in quantification. The quantification procedure is as follows. For

compound P,

(i) identify the chemical group of all the carbons (CH1 or CH2 or CH3),

(ii) find the mean integration value for each group (I1, I2 and I3),

(iii) normalize I1, I2 and I3 using the average polarization transfer from each

group and find average over all the three (QP), which is proportional to the

concentration of the compound in the mixture,

(iv) repeat the same for all the compounds and find QP, QM and QD. Relative

quantities are the ratios of QP, QM and QD.

Since reduction in experimental time is the major concern, we performed the

GAQIC experiments to obtain minimum experimental time without significant

compromise on the quantification accuracy. The different GAQIC delay sets run

for different number of scans according to their average polarization transfer values.

Case A optimization has the minimum average polarization transfer efficiency (≈45%) and requires the highest number of scans (256) whereas case C optimization

(average polarization transfer ≈ 70%) needs only 128 scans. The experimentally


Delay Set Scans RQ21 RQ31

Q 512 2.42 1.39A8

256

2.33 1.35A8´ 2.35 1.31A16 2.33 1.31H8 2.40 1.35H8´ 2.38 1.30B8

1602.43 1.38

B8´ 2.37 1.35C8 128 2.43 1.37

Table 6.2: Experimentally measured relative quantities of compound 2 (RQ21)(α-methoxy phenyl acetic acid) and compound 3 (RQ22) (1-2-dimethyl cyclo-hexane) with respect to compound 1 (4-penten 2 ol). Relative quantity byweight is shown in Eqn. 6.10.

(a)

(b)

Figure 6.7: Experimental implementation of GAQIC spectral editing delayset. (a). 1H decoupled 13C spectra and (b). using the delay set (S4) shownin Fig. 6.3(a), CH2 peaks (indicated by ↓ mark above) are missing in the fourscan averaged INEPT spectrum.

measured relative quantities (RQ21 and RQ31) are compared with the traditional13C quantitative experiment (Q) in Tab. 6.2. Among all GAQIC delay sets, C

shows the most accurate quantification and needs only half the time compared

to Q-INEPT-CT of Makela et al. T1 and T2 relaxation times of all the com-

pounds are measured and is well within the limits of quantification (T1, T2)= (9.8, 2.5)P , (6.7, 1.7)M , (10.4, 3.9)D seconds. Experimental implementation of

spectral editing using S4 (four scan delay set) is shown in Fig. 6.7. It is seen that

CH2 peaks are almost absent in Fig. 6.7(b) and hence S4 delay set can be used

for spectral simplification and quantification in complex mixtures. Delay set S16

(sixteen scan) should be used for high accuracy of quantification.


Conclusion

Using Genetic algorithm optimization with various fitness functions, we have gen-

erated optimized delay sets, which can perform faster and accurate quantification.

Experimental time advantage of these delay sets are shown by performing GAQIC

experiments in a mixture of α-methoxy phenyl acetic acid, 1-2-dimethyl cyclohex-

ane and 4-penten 2 ol. We also discussed quantitative spectral editing technique

(spectral simplification and quantification) for complex mixtures.

Appendix A

Unitary Decomposition of

Quantum Gates Using GA

Optimization

Here we discuss optimal experimental unitary decomposition of some of the impor-

tant quantum gates as well as state preparations using Genetic Algorithm. The

theory of UOD using GA is explained in Chapter 2.

The representation format of all the UODs in this appendix is as follows,

(i) θ(φ)n represents a θ length pulse with phase φ on nth spin (Eqn. A.1),

Rn(θ, φ) = exp(−iθ × [Cosφ Inx + Sinφ Iny ]). (A.1)

(ii) Uzz(Θ) represents spin system evolution under scalar coupling (Eqn. A.2).

Uzz(Θ) = exp(−i Θ 2IzIz). (A.2)

A.1 Controlled Hadamard (C-H)

Controlled Hadamard is one of the basic two qubit controlled operation [128].

The action of the gate is as follows - “If the Control qubit state is |1〉, apply

103

Appendix A. Quantum Gates using GA Optimization 104

Figure A.1: Pulse sequence for Controlled Hadamard.

a Hadamard operation on the target qubit.”. The UOD of C-H using GA is an

operator optimization (discussed in Chapter 2) with target unitary (Eqn. A.3),

C-H =

1 0 0 0

0 1 0 0

0 0 1/√

2 −1/√

2

0 0 1/√

2 1/√

2

. (A.3)

The optimized decomposition / pulse sequence is obtained as (Eqn. A.4 & Fig.

A.1),

C-H = 135(90)2 : Uzz(90) : 128.63(143.10)2 :

180(135)1 : 90(118)2.(A.4)

The pulse sequence has four SQRs and single 1/2J evolution under spin scalar

coupling.

A.2 Controlled-Controlled-NOT (CCNOT) Gate

Controlled-Controlled-NOT or Toffoli gate (Eqn. A.5) [128, 129] is a three qubit

universal reversible gate proposed by Tommaso Toffoli in 1980. Toffoli gate has two

control qubits and a target qubit. The action of Toffoli gate can be defined as (Eqn.

A.5) - “If the two control qubits are at |11〉, apply ‘NOT’ operation to the target”.

The variants of Toffoli gate based on the different states – |00〉, |01〉, |10〉, |11〉– of the control qubits are CCNOT(1,2) 3, CCNOT(1,2) 3, CCNOT(1,2) 3,


Figure A.2: Pulse sequence for Toffoli Gate.

Operator φ1 φ2 φ3 φ4 φ5

CCNOT(1,2) 3 3π/8 3π/4 7π/8 π/4 π/2CCNOT(1,2) 3 5π/8 5π/4 15π/8 3π/4 π/2CCNOT(1,2) 3 11π/8 3π/4 π/8 5π/4 3π/2CCNOT(1,2) 3 π/2 π/4 π/8 3π/4 3π/2

Table A.1: Phase modulation values for different Toffoli gates

CCNOT(1,2) 3.

CCNOT(1,2) 3 =

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0

. (A.5)

Using GA optimization, we have obtained the CCNOT gate decomposition and is

shown in Fig. A.2. Different variants can be obtained by modulating the phases

φi=1...5 as shown in Tab. A.1.

A.3 Controlled SWAP (C-S)

Controlled SWAP gate or Fredkin gate (Eqn. A.3) [128, 129] is a three qubit

reversible operation invented by Ed Fredkin. The action of Controlled SWAP


Figure A.3: Controlled-SWAP decomposition using C-NOT gates.

(Eqn. A.6) is as follows - “If the control qubit state is |1〉, SWAP (Eqn. A.7) the

states of two target qubits.”.

C-S =

[I4×4 0

0 USWAP

], (A.6)

where I4×4 is the identity matrix of dimension 4 and

USWAP =

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

(A.7)

Using the C-NOT decomposition of SWAP gate (Fig. A.3), Controlled- SWAP

can be built using three Controlled-Controlled-NOT gates. The heavy resource

requirement (3×CCNOT; Fig. A.2) of this decomposition can lead to experimental

errors. The optimized decomposition using GA is obtained as(Eqn. A.8 & Fig.

A.4) ,

C-S = 90(90)2 : U13zz (90) : 180(22.5)1 : U23

zz (90) : 90(180)2 :

45(0)3 : U13zz (90) : U23

zz (90) : U12zz (90) : 90(90)3 : U23

zz (45) :

U13zz (45) : 90(90)3 : 180(112.5)2 : 90(270)2 : 45(180)3 :

U23zz (45) : 90(180)2 : 180(0)1.

(A.8)

A.4 Quantum Counter circuits

Quantum counter operator transforms an nth quantum state in the computational

basis to (n+ s)th state, where s is the step size. The basic counter circuit for an n

qubit can be made using Controlled NOT Gates – three qubit example is shown

in Fig. A.5.


Figure A.4: Pulse sequence for Controlled SWAP gate.

Figure A.5: Three qubit quantum counter using CC-NOT and C-NOT Gates.

Initial State Final State|00〉 |01〉|01〉 |10〉|10〉 |11〉|11〉 |00〉

Table A.2: Truth table for two qubit quantum counter.

The action of two qubit counter operator with step 1 (U2C1) is shown in Tab. A.2.

The operator matrix for two qubit quantum counter (Tab. A.2) is given by,

U2C1 =

0 0 0 1

1 0 0 0

0 1 0 0

0 0 1 0

The optimized unitary decomposition for the two qubit counter is obtained as

(Eqn. A.9),

U2C1 = 180(45)2 : 90(90)1 : Uzz(90) : 90(0)1 : 180(45)1. (A.9)


Figure A.6: Pulse sequence for Two Qubit Counter.

The pulse sequence (Fig. A.6) has four SQRs and a single delay of length 1/2J .

The operator matrix for three qubit quantum counter (U3C1 – counting one at a

time) is given by (Eqn.A.10),

U3C1 =

0 0 0 0 0 0 0 1

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

(A.10)

The optimized decomposition is obtained as (Eqn. A.11),

U3C1 = 90(90)1 : U12zz (45) : 90(45)2 : U13

zz (45) : U23zz (90) :

180(112.5)1 : 180(90)2 : 90(45)3 : 180(112.5)3 :

U12zz (45) : 90(90)3.

(A.11)

The pulse sequence (Fig. A.7) has seven SQRs and four scalar coupling evolutions.

The decomposition using Controlled NOT gates as shown in Fig. A.5 requires more

resources.

Counting two at time (U3C2) can be easily implemented using the following rela-

tion,

U3C2 = (U2C1 ⊗ I2×2), (A.12)


Figure A.7: Pulse sequence for Three Qubit Counter.

where I2×2 is the identity matrix and U2C1 is given in Eqn. A.9. Likewise counting

three at a time can implemented by,

U3C3 = U3C1 × (U2C1 ⊗ I2×2). (A.13)

A.5 W State Creation

W state is a three qubit maximally entangled state [128] and is given by (Eqn.A.14),

|W 〉 =1√3

(|001〉+ |101〉+ |100〉). (A.14)

W state is robust against particle loss error and is a good candidate for ensemble

based quantum memories [130].

Starting from |000〉 state, the optimized unitary decomposition for W state cre-

ation using GA is obtained as (Eqn. A.15),

UW = U23zz (90) : U12

zz (90) : 90(0)1 : 90(180)2 : U23zz (45) :

U12zz (90) : 90(90)2 : 70.53(90)3 : 90(315)2 : 90(90)1.

(A.15)

The pulse sequence (Fig. A.8) has six SQRs and four neighbor scalar coupling

evolutions.


Figure A.8: Pulse sequence for W state creation.

Figure A.9: Quantum circuit for probability splitter.

Figure A.10: Pulse sequence for Quantum Probability Splitter.

A.6 Quantum Probability Splitter (QPS)

Quantum Probability Splitter (QPS) modifies the probability amplitudes of a qubit

so that the probability on the selected basis state is halved [131]. Quantum state

creation is the potential application of QPS. The circuit for quantum probability

splitter is shown in Fig. A.9 [131].


The optimized decomposition using GA is obtained as (A.16),

UQPS = 90(45)2 : U23zz (90) : 90(315)2 : 90(225)3 : U12

zz (45) :

U23zz (90) : U13

zz (90) : 60(22.5)1 : 225(45)2 : 90(45)3 :

U23zz (90) : 120(22.5)1 : 135(135)2 : U12

zz (90) :

U23zz (135) : 180(0)1.

(A.16)

The pulse sequence (Fig. A.10) uses nine SQRs and six scalar coupling evolution

delays.

Appendix B

Matlab UOD Package

Experimental implementation of a quantum gate or an unitary operator (U) re-

quires decomposition of U , in terms of experimentally available system unitaries

– such as rf pulses and evolution of zz interaction (in NMR). This decomposition

should be optimal in terms of available resources to reduce experimental errors.

A Pulse sequence in Nuclear Magnetic Resonance can be considered as a decom-

posed unitary representation, where main unitary (U) is given by the equivalent

average Hamiltonian. Unitary operator decomposition or Pulse sequence gener-

ation of a unitary operator, U can be treated as an optimization problem by

defining a function (F) in such a way that the function profile of F defined by

pulse sequence parameters must have a minimum or maximum point for best pos-

sible pulse sequence or unitary decomposition of U . Here using Genetic Algorithm

optimization, we describe a scheme for generating optimal pulse sequence or uni-

tary decomposition for any unitary operator. Genetic algorithm optimization is a

global search method inspired by nature’s evolutionary process [34]. Genetic al-

gorithm has been used in NMR for designing new experiments [40], for improving

excitation and inversion accuracy of rf pulses [41] and pulse sequence optimization

[42].

The details of GA optimization such as representation scheme and fitness function

are defined in Chapter 2. The Matlab package for Unitary Operator Decomposition

(UOD) is explained here with a typical example of three qubit target operator -

Controlled SWAP (C-S). Controlled SWAP applies SWAP operation between two

113

Appendix B. Matlab UOD package 114

target qubits if the control qubit is in |1〉 state (Eqn. B.1).

Target operator, C-S =

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

(B.1)

This target operator can be replaced by any three qubit unitary operator for

performing its UOD. The output of the program should be decoded using the

representation scheme (explained in Chapter 2). The program package given here

is a three qubit variant of UOD. The generalization of the program to any qubit

is in development stage. The UOD Matlab package has 11 subroutines working

together with Matlab’s optimization toolbox optimtool [132, 133]. All programs

are developed in Matlab® 7.14, 32 bit Ubuntu 12.04 OS with core i5 3.6 GHz

Intel processor.

UOD optimization package performs multiple GA optimizations in a loop with

population size and number of generations are defined in uod.m. The convergence

checking of the optimization and applying constraints on the population are out-

side these GA segments. Matlab subroutine list used in UOD package is explained

below,

1. uod.m: This is the main user interface program. Target operator and other

problem specific Genetic algorithm optimization details – such as population

size, generation, initial population etc. – are specified here.

Problem specific variables used in ‘uod.m’ are,

utar : Target unitary operator.

utarName : Name of the target unitary operator.

Population-

Size Data

: number of individuals in the population.

Generation-

s Data

: Number of generations.


segno : number of segments or basic unit of pulse sequence (see

Chapter 2).

ctoffx : used for checking the convergence and termination. If the

fitness value difference between two consecutive run is less

than ctoffx, the program will count it into the termination

counter.

lpno : number of GA optimization segments.

stopat : Maximum fitness value (in %) to be satisfied for basic op-

timizations. Refining the solution segment of the program

will make near 100% fidelity.

initialmutaion-

ratio

: Initial value for mutation ratio. This value will change

according to the convergence during the Optimization.

initial-

population

: Specify the initial population here. GA will generate a

random population if initialpopulation field is blank.

EliteCount-

Number

: Specifies the number of elite individuals. Elite generation

is explained in Chapter 1.

MutationSwit-

ching flag

: Switch the subroutine MutationSwitching on. This will

change the basic mutation function of basic GA optimiza-

tion alternatively between uniform and adaptivefeasible

inside main loop.

ApplyGaxAppr-

oximations flag

: Switch the subroutine ApplyGaxApproximations on. This

will round each pulse parameters in the individual with

0, π/2, π.

GaxKicking-

flag

: Switch the subroutine GaxKicking on. This applies con-

stant perturbation to the final individual (gax), by scaling

up or down. ApplyGaxApproximations and GaxKicking

helps to keep the optimization away from local minimas.

2. uodmain.m: The main program which controls the optimization flow. Set-

ting up of initial population opening new data file for logging the optimiza-

tion are performed here. The basic Optimization flow is as follows,

(i) Direct initial optimization with all resources available. This usually

results in a pulse sequence with more number of delays and pulses than

optimal value required.


(ii) ‘Delay Zeroing ’ procedure: to reduce the length of the pulse sequence

obtained from Step.(i).

(iii) ‘Pulse Zeroing ’ procedure: to reduce the number of pulses in the pulse

sequence obtained from Step.(ii).

(iv) Final refining of the solution obtained from Step. (iii). This will make

the fidelity near to 100%

3. SpinOperators.m: Basic spin product operators defined here.

4. GADefaults.m: Sets the default values of Genetic Algorithm optimization.

The main default values set in UOD are,

(i) ‘StallGenLimit Data’ is kept as number of generations. (The algorithm

stops if the weighted average relative change in the best fitness function

value over Stall generations is less than or equal to Function tolerance).

(ii) ‘TolFun Data=0’. (The algorithm stops if the weighted average relative

change in the best fitness function value over Stall generations is less

than or equal to Function tolerance[133]).

(iii) TolCon Data=0. (It is used to determine the feasibility with respect to

nonlinear constraints [133]).

(iv) Cross over function is set as crossoverheuristic (explained in Chapter

1).

(v) Parental selection function is set as selectionroulette (explained in Chap-

ter 1).

5. uodFitness.m: Fitness function of the GA optimization. The fitness function

used for UOD is explained in Chapter 2.

6. uodFitnessPulseO.m: The fitness function used during pulse zeroing.

7. ApplyGaxConstraints.m: Sets the conditions on pulse parameters such as

maximum value of delay length and rotation angle of pulse.

8. optimize delay.m: Subroutine for delay zeroing – to reduce the pulse se-

quence length.

9. MutationSwitching.m: This change the basic mutation function of GA opti-

mizations inside the main loop alternatively between uniform and adaptive-

feasible(Chapter 1). This action found to be converging faster than single

mutation function.


10. GaxKicking.m: Applies a small perturbation in the form of scaling up or

down or adding small random number to best individual in the population.

This action helps to keep optimization out of local minimas.

11. uniform.m: Applies the mutation function uniform to the population. Func-

tion uniform is explained in Chapter 1.

The individual pulse sequences has ‘segno’ of pulse-delay segments, which makes

it ‘9 × segno’ number of pulse sequence parameters and is represented by an

array ‘gax ’ (as given in matlab code ‘uodFitness.m’). The first 6 × segno of gax

elements represents the pulse parameters (odd numbers pulse angles and even

numbers pulses phases) and remaining elements represents the delay segments,

which is again devided into groups of three elements with first element in a group

represents the coupling evolution of spins 1 and 2, second element represents the

coupling evolution of spins 2 and 3 and third element represents coupling evolution

of spins 1 and 3.


uod.m

c l e a r a l l2 c l c

g l o b a l segno utar4 g l o b a l i n i t i a l m u t a i o n r a t i o dt in

g l o b a l Populat ionSize Data Generat ions Data c t o f f x lpno6 g l o b a l r e s e t count s topat i n i t i a l p o p u l a t i o n EliteCountNumber

utarNameg l o b a l ApplyGaxApproximations f lag Mutat ionSwi tch ing f l ag

GaxKick ing f lag8 g l o b a l x y z I I

t i c10 x=[0 1 ;1 0 ] / 2 ; y=[ 0 −1 i ; 1 i 0 ] / 2 ; z =[1 0 ; 0 −1]/2; I I=eye (2 )

;

12 % Def ine Target operator −− utarutarName = ’ cswap ’ ; % name o f gate

14 utar = [ . . .1 0 0 0 0 0 0 0

16 0 1 0 0 0 0 0 00 0 1 0 0 0 0 0

18 0 0 0 1 0 0 0 00 0 0 0 1 0 0 0

20 0 0 0 0 0 0 1 00 0 0 0 0 1 0 0

22 0 0 0 0 0 0 0 1 ] ;

24 % D e t a i l s o f Genet ic Algorithm ParametersPopulat ionSize Data =40;

26 Generat ions Data =60;segno =9; % number o f segments in pu l s e sequence

28 c t o f f x =0.0125; % Zero approximation c u t o f f f o r parameter ini n d i v i d u a l s

lpno =999999; % I t e r a t i o n s no o f above (Populat ionSize Data , Generat ions Data )

30 r e s e t count =5; % Reset to i n i t i a l i f f i t n e s s cont inues f o rr e s e t count t imes

s topat =96.0 ;32 i n i t i a l m u t a i o n r a t i o =0.11;

i n i t i a l p o p u l a t i o n=pi /2* ones (1 , segno *9) ;34 EliteCountNumber=2;

Mutat ionSwi tch ing f l ag = 1 ;36 ApplyGaxApproximations f lag = 1 ;

GaxKick ing f lag =1;38 dt in = d a t e s t r (now , ’mmmm dd , yyyy HH:MM: SS .FFF AM’ ) ;

d i sp ( [ ’ Optimizat ion Started : ’ dt in ] ) ;40 di sp ( ’ Target Unitary Operator : ’ ) ;

d i sp ( num2str ( utar ) ) ;42 di sp ( [ ’ Populat ion S i z e : ’ num2str ( Populat ionSize Data ) ] ) ;


di sp ( [ ’ Generat ions : ’ num2str ( Generat ions Data ) ] ) ;44 di sp ( [ ’ E l i t e Count : ’ num2str ( EliteCountNumber ) ] ) ;

i f Mutat ionSwi tch ing f l ag ; d i sp ( ’ Mutation Switching ON’ ) ; end ;46 i f ApplyGaxApproximations f lag ; d i sp ( ’Gax Approximations ON’ ) ;

end ;uodmain ; % main program with d e f a u l t va lue s o f GA

opt imiza t i on48 dt = d a t e s t r (now , ’mmmm dd , yyyy HH:MM: SS .FFF AM’ ) ;

toc

uodmain.m

f unc t i on [ gax , f v a l ] = uodmain2 g l o b a l segno utar utarName

g l o b a l lpno r e s e t count s topat opt ions4 g l o b a l d l z f l g gaxsave i d x d l y z e r o s idxd lynonzeros d lyar ray

i n i t i a l p o p u l a t i o ng l o b a l ApplyGaxApproximations f lag Mutat ionSwi tch ing f l ag

GaxKick ing f lag6 g l o b a l p l z f l g p l s a r r a y d e l a y s t a r t delayend dt in i d x p l s z e r o s

g a x f i n a l delaysum

8 % c r e a t i n g d a t a f i l e f o r output wr i t i ngf i l e I D = fopen ( [ utarName ’ . txt ’ ] , ’ a ’ ) ;

10

f p r i n t f ( f i l e I D , [ ’ \n\n−−−−−−−−New Optimisation−−−−−−−−−−\nStarted On: ’ dt in ’ \n ’ ] ) ;

12

SpinOperators % Three sp in Product ope ra to r s14

utar=utar ’ ; % us ing in f i t n e s s eva lua t i on16 nvars=9*segno ;

18 f v a l 1 =0;d e l a y s t a r t =6*( segno ) +1; delayend=nvars ;

20 dlyar ray=d e l a y s t a r t : delayend ;p l s a r r a y =1:2 : de l ay s ta r t −1;

22

GADefaults % Defau l t va lue s f o r GA parameters24

% I n i t i a l Populat ion S e t t i n g s . . . . . . .26 i f isempty ( i n i t i a l p o p u l a t i o n )

opt ions = gaopt imset ( opt ions , ’ I n i t i a l P o p u l a t i o n ’ , 2* pi *rand (1 , nvars ) ) ; gax=2*pi * rand (1 , nvars ) ;

28 e l s eopt ions = gaopt imset ( opt ions , ’ I n i t i a l P o p u l a t i o n ’ ,

i n i t i a l p o p u l a t i o n ) ; gax=i n i t i a l p o p u l a t i o n ;30 end


g a x f i n a l=gax ;32 r e s e t chk ( 1 : 1 0 0 ) =0;

d l z f l g =0; endde layze ro ing =0; endpu l s e z e ro ing =0; p l z f l g =0;34 f p r i n t f ( [ ’ \n\nStep (/ ’ num2str ( lpno ) ’ ) \ t F i tne s s \n ’ ] ) ;

f p r i n t f ( ’−−−−−−−−−−−−−−−−−−−− \n ’ ) ;36 f o r cnt =1: lpno

i f endde layzero ing && endpu l s e z e ro ing38 break

e l s e40 [ gax , f va l , ˜ , ˜ , ˜ , ˜ ] = . . .

ga ( @uodFitness , nvars , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , opt i ons ) ;42 gax=ApplyGaxConstraints ( gax ) ;

f v a l =12.5* abs ( f v a l ) ;44 f p r i n t f ( [ num2str ( cnt ) ’ \ t \ t \ t ’ num2str ( f v a l ) ’ \n ’ ] ) ;

f p r i n t f ( f i l e I D , ’ \nloop no : %d/%d , f i t n e s s va lue : %5.2 f’ , cnt , lpno , f v a l ) ;

46 end

48 % Stopping C r i t e r i o ni f f va l> s topat

50 di sp ( [ ’ f i t n e s s va lue : ’ num2str ( f v a l ) ] )f p r i n t f ( f i l e I D , ’ \ nFi tnes s Value : %5.2 f \nBest

I n d i v i d u a l : ’ , f v a l ) ;52 f p r i n t f ( f i l e I D , ’%f ’ , gax ) ;

i f ˜ p l z f l g54 opt im i z e de l ay % f o r z e ro ing o f de lays

end56

i f endde layzero ing58 t imetaken= toc ;

f p r i n t f ( f i l e I D , [ ’ \n\nTime Taken f o rOptimisat ion : ’ num2str ( t imetaken ) ’ Seconds ’ ] ) ;

60 d t f i = d a t e s t r (now , ’mmmm dd , yyyy HH:MM: SS .FFFAM’ ) ;

d i sp ( d t f i )62 f p r i n t f ( f i l e I D , [ ’ \nFinished at : ’ d t f i ’ \n

−−−−−−−−End o f Optimisation−−−−−−−−−−\n ’ ] ) ;break

64 endend

66

i f ApplyGaxApproximations f lag68 gax=ApplyGaxApproximations ( gax ) ; % Aproximations f o r 0 ,

p i /2 and pi parametersend

70

% Checking Reset c o n d i t i o n s72 i f abs ( fva l1−f v a l ) < 1

f i d e l i t y n o=round ( f v a l ) ;


74 r e s e t chk ( f i d e l i t y n o )=re s e t chk ( f i d e l i t y n o ) +1;i f r e s e t chk ( f i d e l i t y n o ) > ( r e s e t count ) | | f v a l > s topat

76 i f ( d l z f l g && re s e t chk ( f i d e l i t y n o ) > 2* c e i l ( f v a l/15) ) | | f v a l > s topat % ” | | f v a l > s topat ” i s nece s sa ryf o r the case o f s p e c i f i e d f u l l s e t o f i d x d l y z e r o s andidxd lynonzeros

i f f v a l >= stopat78 di sp ( ’ Delay ze ro ing cont inu ing . ! ! ! ’ ) ;

gax=gaxsave ;80 g a x f i n a l=gax ;

e l s e82 di sp ( ’ Delay ze ro ing f a i l e d ! ! ! . ’ ) ;

idxd lynonzeros =[ idxd lynonzeros i d x d l y z e r o s (l ength ( i d x d l y z e r o s ) ) ] ;

84 gax=gaxsave ; i d x d l y z e r o s=i d x d l y z e r o s ( 1 :l ength ( i d x d l y z e r o s )−1) ;

end86

i f l ength ( [ i d x d l y z e r o s idxd lynonzeros ] ) ==delayend − d e l a y s t a r t + 1

88 d i s p l a y ( [ ’ F i d e l i t y : ’ num2str ( abs (uodFitness ( gax ) ) *12 .5 ) ] ) ;

gax ( i d x d l y z e r o s )=ze ro s (1 , l ength ( i d x d l y z e r o s) ) ;

90 d i s p l a y ( [ ’ Zeroed i n d i c e s = ’ num2str (i d x d l y z e r o s ) ] )

endde layze ro ing =1;92 p l z f l g =1;

end94

e l s e i f (˜ d l z f l g && f i d e l i t y n o >= 89 && res e t chk (f i d e l i t y n o ) > ( r e s e t count +3*( f i d e l i t y n o −89)+0* d l z f l g *5) ) | |( ˜ d l z f l g && f i d e l i t y n o < 89 && re s e t chk ( f i d e l i t y n o ) >r e s e t count ) % ”+ d l z f l g *10” added f o r g i v i ng morei t e r a t i o n s in z e ro ing proce s s . . .

96 di sp ( ’ r e s e t i n g . . . . . . ’ ) ;f p r i n t f ( f i l e I D , ’ \nReset ing . . . . . \ n Best

I n d i v i d u a l : ’ ) ;98 f p r i n t f ( f i l e I D , ’%f ’ , gax ) ;

gax=pi * rand (1 , nvars ) ; r e s e t chk ( 1 : 1 0 0 ) =0;d l z f l g =0;

100 endend

102

% a l t e r n a t e l y g i v ing d i f f e r e n t mutation func t i on with d i f f e r e n tparameters

104 i f Mutat ionSwi tch ing f l agMutationSwitching ( fva l , cnt )

106 end


108 % f o r s t a b l e i n d i v i d u a li f GaxKick ing f lag

110 gax=GaxKicking ( gax , f v a l ) ;end

112

e l s e i f f v a l 1 > f v a l114 di sp ( ’NOT A STABLE POINT . . . . Ro l l i ng back with Mutation

Guassian . . . . . . ’ )opt i ons = gaopt imset ( opt ions , ’ MutationFcn ’ ,

@mutationgaussian 1 .23 1 ) ;116 end

118 f v a l 1=f v a l ;opt i ons = gaoptimset ( opt ions , ’ I n i t i a l P o p u l a t i o n ’ , gax ) ;

120 endgax=g a x f i n a l ;

122 g a x f i n a l = ApplyGaxConstraints ( g a x f i n a l ) ;f p r i n t f ( [ ’DELAY ZEROING FINISHED . . . . . . . . . . \n Fina l F i d e l i t y

After Delay Zero ing i s : ’ num2str (−12.5* uodFitness ( g a x f i n a l )) ] ) ;

124 f p r i n t f ( [ ’ \n Fina l GAX i s : ’ num2str ( g a x f i n a l ) ] ) ;

126 delaysum = sum( g a x f i n a l ( d e l a y s t a r t : delayend ) ) ;

128 f p r i n t f ( ’ \n\n\n\n\n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n Pulse Zero ing Started . . . . \ n ’ ) ;

c n t p l s =1;130 numof i ter =0; f l g p l s c h a n g e =1;

i d x p l s z e r o s = [ ] ; numof i ter ( 1 : 1 0 0 ) =0;132 opt ions = gaopt imset ( opt ions , ’ MutationFcn ’ ,

@mutat ionadapt feas ib le ) ;f o r k=cnt : lpno

134 i f f l g p l s c h a n g ed i sp ( [ ’ Zeroed Pulse I n d i c e s : ’ num2str ( i d x p l s z e r o s ) ] ) ;

136

% S e l e c t i n g best pu l s e parameter from p l s a r r a y to makeze ro ing

138 PLS=[ g a x f i n a l ( p l s a r r a y ) ; p l s a r r a y ] ;PLS=sort rows (PLS . ’ , 1 ) . ’ ;

140 p l s a r r a y=PLS ( 2 , : ) ;i d x p l s z e r o s =[ i d x p l s z e r o s p l s a r r a y ( c n t p l s ) ] ;

142

c n t p l s=c n t p l s +1; f l g p l s c h a n g e =0;144 end

[ gax , fva l , ˜ , ˜ , ˜ , ˜ ] = . . .146 ga ( @uodFitnessPulseO , nvars , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] ,

opt i ons ) ;gax=ApplyGaxConstraints ( gax ) ;

148 f v a l =12.5* abs ( f v a l ) ;numof i ter ( round ( f v a l ) )=numof i ter ( round ( f v a l ) ) +1;


150 f p r i n t f ( [ num2str ( k ) ’ \ t \ t \ t ’ num2str ( f v a l ) ’ \n ’ ] ) ;f p r i n t f ( f i l e I D , ’ \nloop no : %d/%d , f i t n e s s va lue : %5.2 f

’ , cnt , lpno , f v a l ) ;152 opt ions = gaopt imset ( opt ions , ’ I n i t i a l P o p u l a t i o n ’ , gax ) ;

i f f v a l > 97 % stopat154 g a x f i n a l=gax ;

opt ions = gaopt imset ( opt ions , ’ I n i t i a l P o p u l a t i o n ’ ,g a x f i n a l ) ;

156 f l g p l s c h a n g e =1;numof i ter ( 1 : 1 0 0 ) =0;

158 i f c n t p l s > l ength ( p l s a r r a y ) ; break ; ende l s e

160 i f numof i ter ( round ( f v a l ) ) > 5f l g p l s c h a n g e =1;

162 opt ions = gaopt imset ( opt ions , ’ I n i t i a l P o p u l a t i o n ’ ,g a x f i n a l ) ;

d i sp ( ’ Pulse z e ro ing f a i l e d ! ! ! ’ ) ;164 i d x p l s z e r o s=i d x p l s z e r o s ( 1 : end−1) ;

numof i ter ( 1 : 1 0 0 ) =0;166 i f c n t p l s > l ength ( p l s a r r a y ) ; break ; end

end168 end

end170 i d x p l s z e r o s=s o r t ( [ i d x p l s z e r o s i d x p l s z e r o s +1]) ;

g a x f i n a l = ApplyGaxConstraints ( g a x f i n a l ) ;172

%%%%%%%%%% FINAL REFINING THE SOLUTION %%%%%%%%%%%%%%174 di sp ( ’ **** REFINING THE SOLUTION **** ’ )

c1=k ; c2 =1;176 whi le ( f v a l < 9 8 . 5 ) && ( c2 < 100)

[ gax , fva l , ˜ , ˜ , ˜ , ˜ ] = . . .178 ga ( @uodFitness , nvars , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , opt ions ) ;

gax=ApplyGaxConstraints ( gax ) ;180 f v a l =12.5* abs ( f v a l ) ;

f p r i n t f ( [ num2str ( c1 ) ’ \ t \ t \ t ’ num2str ( f v a l ) ’ \n ’ ] ) ;182 f p r i n t f ( f i l e I D , ’ \nloop no : %d/%d , f i t n e s s va lue : %5.2 f

’ , cnt , lpno , f v a l ) ;opt i ons = gaopt imset ( opt ions , ’ I n i t i a l P o p u l a t i o n ’ , gax ) ;

184 c1=c1 +1; c2=c2 +1;end

186

%%%%%%%%%%% PRINTING THE SOLUTION %%%%%%%%%%%%%%188 g a x f i n a l = ApplyGaxConstraints ( gax ) ;

d i s p l a y ( [ ’ Zeroed p u l s e s are : ’ num2str ( i d x p l s z e r o s ) ] ) ;190 d i s p l a y ( [ ’ F ina l F i d e l i t y i s : ’ num2str ( abs ( uodFitness ( g a x f i n a l

) *12 .5 ) ) ] ) ;d i s p l a y ( [ ’ F ina l GAX i s : ’ num2str ( g a x f i n a l ) ] ) ;


SpinOperators.m

f unc t i on SpinOperators2 g l o b a l x y z I I x i i y i i i x i i y i i i x i i y z z i i z z z i z z i i i z i

i i zx i i=mkron (x , I I , I I ) ; y i i=mkron (y , I I , I I ) ;

4 i x i=mkron ( I I , x , I I ) ; i y i=mkron ( I I , y , I I ) ;i i x=mkron ( I I , I I , x ) ; i i y=mkron ( I I , I I , y ) ;

6 z i i=diag (mkron ( z , I I , I I ) ) ; i z i=diag (mkron ( I I , z , I I ) ) ; i i z=diag (mkron ( I I , I I , z ) ) ;

z z i=diag (mkron ( z , z , I I ) ) ; z i z=diag (mkron ( z , I I , z ) ) ; i z z=diag (mkron ( I I , z , z ) ) ;

8 end

GADefaults.m

1 f unc t i on GADefaultsg l o b a l opt ions

3 g l o b a l Populat ionSize Data Generat ions Datag l o b a l i n i t i a l m u t a i o n r a t i o EliteCountNumber

5 Stal lGenLimit Data=Generat ions Data ;TolFun Data=0;TolCon Data=0; opt ions = gaopt imset ;

7 opt ions = gaopt imset ( opt ions , ’ TolFun ’ , TolFun Data ) ;opt i ons = gaopt imset ( opt ions , ’ TolCon ’ , TolCon Data ) ;

9 opt ions = gaopt imset ( opt ions , ’ CrossoverFcn ’ , @ c r o s s o v e r h e u r i s t i c [ ] ) ;

opt i ons = gaopt imset ( opt ions , ’ Popu lat ionS ize ’ ,Populat ionSize Data ) ;

11 opt ions = gaopt imset ( opt ions , ’ Generat ions ’ ,Generat ions Data ) ;

opt i ons = gaopt imset ( opt ions , ’ Sta l lGenLimit ’ ,Stal lGenLimit Data ) ;

13 opt ions = gaopt imset ( opt ions , ’ Se l e c t i onFcn ’ ,@ s e l e c t i o n r o u l e t t e ) ;opt i ons = gaopt imset ( opt ions , ’ Display ’ , ’ o f f ’ ) ;

15 opt ions = gaopt imset ( opt ions , ’ El i teCount ’ , EliteCountNumber) ;

opt i ons = gaopt imset ( opt ions , ’ Vector i zed ’ , ’ o f f ’ ) ;17 opt ions = gaopt imset ( opt ions , ’ Us eP ar a l l e l ’ , ’ always ’ ) ;

opt i ons = gaopt imset ( opt ions , ’ MutationFcn ’ ,@mutat ionadapt feas ib le ) ;

19 opt ions = gaopt imset ( opt ions , ’ El i teCount ’ , EliteCountNumber) ;

end


uodFitness.m

1 f unc t i on f t n s = uodFitness ( gax )g l o b a l segno utar

3 g l o b a l x i i y i i i x i i y i i i x i i y z z i i z z z i z %z i i i z i i i zgax=ApplyGaxConstraints ( gax ) ;

5 u e f f=eye (8 ) ;f o r j =0: segno−1

7 count1=j *6 ;RfMat=−1i *( gax ( count1+1)*( cos ( gax ( count1+2) ) * x i i+s i n ( gax (

count1+2) ) * y i i ) + . . .9 gax ( count1+3)*( cos ( gax ( count1+4) ) * i x i+s i n ( gax ( count1+4) ) * i y i )

+ . . .gax ( count1+5)*( cos ( gax ( count1+6) ) * i i x+s i n ( gax ( count1+6) ) * i i y ) ) ;

11 [ v1 , d1]= e i g (RfMat) ;cc =6*( segno )+3* j ;

13 ZMat=exp(−1 i *( gax ( cc+1)* z z i+gax ( cc+2)* i z z+gax ( cc+3)* z i z ) ) ;u e f f =( u e f f *v1*diag ( exp ( diag ( d1 ) ) ) /v1 ) *diag (ZMat) ;

15 endf t n s=−(abs ( t r a c e ( utar * u e f f ) ) ) ; % utar here i s uatr ’

uodFitnessPulseO.m

f unc t i on f t n s = uodFitnessPulseO ( gax )2 g l o b a l segno utar d e l a y s t a r t delayend delaysum

g l o b a l x i i y i i i x i i y i i i x i i y z z i i z z z i z %z i i i z i i i z4 gax=ApplyGaxConstraints ( gax ) ;

u e f f=eye (8 ) ;6 f o r j =0: segno−1

count1=j *6 ;8 RfMat=−1i *( gax ( count1+1)*( cos ( gax ( count1+2) ) * x i i+s i n ( gax (

count1+2) ) * y i i ) + . . .gax ( count1+3)*( cos ( gax ( count1+4) ) * i x i+s i n ( gax ( count1+4) ) * i y i )

+ . . .10 gax ( count1+5)*( cos ( gax ( count1+6) ) * i i x+s i n ( gax ( count1+6) ) * i i y ) ) ;

[ v1 , d1]= e i g (RfMat) ;12 cc =6*( segno )+3* j ;

ZMat=exp(−1 i *( gax ( cc+1)* z z i+gax ( cc+2)* i z z+gax ( cc+3)* z i z ) ) ;14 u e f f =( u e f f *v1*diag ( exp ( diag ( d1 ) ) ) /v1 ) *diag (ZMat) ;

end16 i f (sum( gax ( d e l a y s t a r t : delayend ) ) ) <= ( delaysum+1)

f t n s=−(abs ( t r a c e ( utar * u e f f ) ) ) ;18 e l s e

f t n s =−1;20 end


ApplyGaxConstraints.m

2 f unc t i on [ gaxout ] = ApplyGaxConstraints ( gaxin )g l o b a l d e l a y s t a r t delayend i d x d l y z e r o s i d x p l s z e r o s

4 gaxout ( 1 : ( de l ay s ta r t −1) )=mod( gaxin ( 1 : ( de l ay s ta r t −1) ) ,2* pi ) ;gaxout ( d e l a y s t a r t : delayend )=mod( gaxin ( d e l a y s t a r t : delayend ) , p i ) ;

6 gaxout ( [ i d x d l y z e r o s i d x p l s z e r o s ] ) =0;end

optimize delay.m

2 g l o b a l i d x d l y z e r o s d l z f l g idxd lynonzeros d lyar rayc l e a r z e ro idx gaxsave

4 ze ro idx = [ ] ;gaxsave=gax ;

6 l ngz ro=length ( d lyar ray ) ;f zp =0;

8 memberchk=˜ismember ( dlyarray , [ i d x d l y z e r o s idxd lynonzeros ] ) ;f o r k=1: lngz ro

10 i f memberchk ( k )g a x t r i a l=gax ;

12 g a x t r i a l ( d lyar ray ( k ) ) =0;f z=abs ( uodFitness ( g a x t r i a l ) ) ;

14 i f f z > f zpfzp=f z ;

16 ze ro idx=dlyar ray ( k ) ;end

18 endend

20 i d x d l y z e r o s =[ i d x d l y z e r o s z e ro idx ] ;d i sp ( ’ Delay Zero ing Started . . . Zeroed I n d i c e s are : ’ ) ;

22 di sp ( i d x d l y z e r o s )d l z f l g =1;

24 r e s e t chk ( 1 : 1 0 0 ) =0;


MutationSwitching.m

1 f unc t i on MutationSwitching ( fva l , cnt )g l o b a l opt ions

3 switch (1+mod( cnt , 2 ) )case 1

5 i f f v a l > 90opt ions = gaopt imset ( opt ions , ’ MutationFcn ’ ,

@uniform (100− f v a l ) /80 ) ;7 e l s e

opt ions = gaopt imset ( opt ions , ’ MutationFcn ’ , @uniform rand (1) /5 ) ;

9 endcase 2

11 opt ions = gaopt imset ( opt ions , ’ MutationFcn ’ ,@mutat ionadapt feas ib le ) ;

end13 end

GaxKicking.m

f unc t i on [ gaxout ] = GaxKicking ( gaxin , f v a l )2 i f f v a l < 95

gaxout=gaxin *0 . 9 7 ;4 e l s e

gaxout=gaxin *0 . 9 9 ;6 end

end


uniform.m

f unc t i on mutationChi ldren = uniform ( parents , opt ions ,GenomeLength , ˜ , ˜ , ˜ , th i sPopu lat ion , mutationRate )

2 mutationChi ldren = ze ro s ( l ength ( parents ) , GenomeLength ) ;f o r i =1: l ength ( parents )

4 c h i l d = th i sPopu la t i on ( parents ( i ) , : ) ;mutationPoints = f i n d ( rand (1 , l ength ( c h i l d ) ) <

mutationRate ) ;6 range = opt ions . PopInitRange ;

[ ˜ , c ] = s i z e ( range ) ;8 i f ( c ˜= 1)

range = range ( : , mutationPoints ) ;10 end

lower = range ( 1 , : ) ;12 upper = range ( 2 , : ) ;

span = upper − lower ;14 c h i l d ( mutationPoints ) = lower + rand (1 , l ength (

mutationPoints ) ) .* span ;mutationChi ldren ( i , : ) = c h i l d ;

16 end


OUTPUT

Optimizat ion Started : September 18 , 2013 1 0 : 0 5 : 4 2 . 1 5 3 PM2 Target Unitary Operator :

1 0 0 0 0 0 0 04 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 06 0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 08 0 0 0 0 0 0 1 0

0 0 0 0 0 1 0 010 0 0 0 0 0 0 0 1

Populat ion S i z e : 4012 Generat ions : 60

E l i t e Count : 214 Mutation Switching ON

Gax Approximations ON16

18 Step (/999999) F i tne s s−−−−−−−−−−−−−−−−−−−−

20 1 66 .3592 69.9054

22 3 72.04964 73.3215

24 5 73 .60714 74 .713

26 15 74.6462r e s e t i n g . . . . . .

28 16 64.583317 72.1024

30 18 73.651324 76.4163

32 25 77 .31626 78.3563

34 29 82.029431 87.4276

36 32 89.628434 93.6552

38 35 95.414236 96.5928

40 f i t n e s s va lue : 96 .5928Delay Zero ing Started . . . Zeroed I n d i c e s are :

42 73

44 37 97.5006f i t n e s s va lue : 97 .5006

46 Delay Zero ing Started . . . Zeroed I n d i c e s are :


73 5748 Delay ze ro ing cont inu ing . ! ! !

38 97 .771150 f i t n e s s va lue : 97 .7711

Delay Zero ing Started . . . Zeroed I n d i c e s are :52 73 57 63

Delay ze ro ing cont inu ing . ! ! !54 39 98.4392

f i t n e s s va lue : 98 .439256 Delay Zero ing Started . . . Zeroed I n d i c e s are :

73 57 63 6658 Delay ze ro ing cont inu ing . ! ! !

40 98 .472660 f i t n e s s va lue : 98 .4726

Delay Zero ing Started . . . Zeroed I n d i c e s are :62 73 57 63 66 75

64 Delay ze ro ing cont inu ing . ! ! !41 98 .7079


68 73 57 63 66 75 74Delay ze ro ing cont inu ing . ! ! !

70 42 98.9116f i t n e s s va lue : 98 .9116

72 Delay Zero ing Started . . . Zeroed I n d i c e s are :73 57 63 66 75 74 61



78 73 57 63 66 75 74 61 78Delay ze ro ing cont inu ing . ! ! !

80 44 99.1897f i t n e s s va lue : 99 .1897

82 Delay Zero ing Started . . . Zeroed I n d i c e s are :73 57 63 66 75 74 61 78 59



88 73 57 63 66 75 74 61 78 59 76Delay ze ro ing cont inu ing . ! ! !

90 46 99.3085f i t n e s s va lue : 99 .3085

92 Delay Zero ing Started . . . Zeroed I n d i c e s are :73 57 63 66 75 74 61 78 59 7679




98 73 57 63 66 75 74 61 78 59 7679 64



73 57 63 66 75 74 61 78 59 7679 64 60



108 73 57 63 66 75 74 61 78 59 7679 64 60 81



114 73 57 63 66 75 74 61 78 59 7679 64 60 81 68



73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55



124 73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69



73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69 67

130 54 97.5145f i t n e s s va lue : 97 .5145

132 Delay Zero ing Started . . . Zeroed I n d i c e s are :73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69 67 80




138 73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69 67 80 71

140 56 98.9964f i t n e s s va lue : 98 .9964

142 Delay Zero ing Started . . . Zeroed I n d i c e s are :73 57 63 66 75 74 61 78 59 76

79 64 60 81 68 55 69 67 80 7162



148 73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69 67 80 7162 72


59 91.8223152 60 91.9674

70 92.3325154 71 92.2263

72 92.2929156 73 92 .191

Delay ze ro ing f a i l e d ! ! ! .158 74 98.9949


73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69 67 80 7162 58

162 75 90.571276 91.4515

164 77 91.777384 92.2361

166 85 92 .12188 92.2745

168 90 92.240391 92.2614

170 Delay ze ro ing f a i l e d ! ! ! .92 99 .4279


174 73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69 67 80 7162 70

176 93 89.130394 90.0468

178 95 90 .407110 92 .133


180 111 92.0422112 92 .145

182 Delay ze ro ing f a i l e d ! ! ! .113 99.5394


186 73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69 67 80 7162 65

114 80.4524188 115 82.9635

116 84.1121190 127 85.0483

128 85.1719192 129 85.0664

Delay ze ro ing f a i l e d ! ! ! .194 130 99.5974


73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69 67 80 7162 77

198 131 77.5263132 78.5236

200 133 78.6551143 78.9574

202 144 78.8042Delay ze ro ing f a i l e d ! ! ! .

204 145 99.7167f i t n e s s va lue : 99 .7167

206 Delay Zero ing Started . . . Zeroed I n d i c e s are :

208 73 57 63 66 75 74 61 78 59 7679 64 60 81 68 55 69 67 80 7162 56

146 82.8367210 147 84.1324

148 84.9284212 158 84.8303

159 85.0946214 160 85.1853

Delay ze ro ing f a i l e d ! ! ! .216 F i d e l i t y : 99 .7167

Zeroed i n d i c e s = 73 57 63 66 75 74 61 78 59 76 79 6460 81 68 55 69 67 80 71 62

218 161 99.7602f i t n e s s va lue : 99 .7602

220 September 18 , 2013 1 0 : 1 3 : 2 0 . 7 1 7 PMDELAY ZEROING FINISHED . . . . . . . . . .

222 Fina l F i d e l i t y After Delay Zero ing i s : 98 .4442


Fina l GAX i s : 0 .052313 1 .9609 0.20021 5 .5731 .4956 1 .7704 6 .2615 3 .0848 1 .1086

0.99784 5 .4168 4 .4772 1 .6643 0.712645 .4304 6 .1377 0.49897 0.80231 0.81889

0.58919 5 .9486 0.67844 0.72385 6 .26896 .2621 0.40098 1 .2504 1 .4099 6 .28124 .1682 0.71421 0.97346 6 .0656 1 .82296 .1095 5 .7344 0.94346 0.92946 0.871985 .4689 0.71925 1 .7229 1 .4546 1 .1623

0.087632 0.13403 5 .4307 5 .1024 0.713191 .1218 0.17422 2 .7155 1 .4414 0.45867

0 2 .8471 0 1 .9202 00 0 0 0 0

2 .709 0 0 0 01 .8733 0 1 .483 0 0

0 0 2 .8545 0 00 0

224

226

228 %%%%%%%%%%%%%%%%%%%%%Pulse Zero ing Started . . . .

230 Zeroed Pulse I n d i c e s :161 99.7847

232 Zeroed Pulse I n d i c e s : 1162 99.8482

234 Zeroed Pulse I n d i c e s : 1 25163 99.8845

236 Zeroed Pulse I n d i c e s : 1 25 7164 99.8804

238 Zeroed Pulse I n d i c e s : 1 25 7 45165 99.6779

240 Zeroed Pulse I n d i c e s : 1 25 7 45 51166 99.7928

242 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3167 98.7937

244 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17168 99.2784

246 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31169 99.5468

248 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29170 98.4426

250 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 41171 98.6473

252 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 4137

172 97.0637254 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 41

37 23173 89.9886


256 174 90 .529178 91.1387

258 179 91 .192Pulse z e ro ing f a i l e d ! ! !

260 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 4137 23

180 94.9882262 181 97.2162

Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 4137 23 19

264 182 89.4439183 90.7552

266 184 91.0245185 91.1873

268 186 91.2343187 91.2565

270 188 91.2705Pulse z e ro ing f a i l e d ! ! !

272 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 4137 23 19

189 90.5005274 190 96.4813

191 97.6454276 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 41

37 23 19 49192 89.9519

278 193 90.8465196 91.1577

280 197 91.1757198 91.1851

282 Pulse z e ro ing f a i l e d ! ! !Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 41

37 23 19 49284 199 83.3913

200 84 .288286 201 84 .797

204 85.2254288 205 85.2591

206 85 .284290 Pulse z e ro ing f a i l e d ! ! !

Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 4137 23 19 49

292 207 78.3271208 78.8203

294 209 78.9421211 79.0401

296 212 79.0447213 79.0495


37 23 19 49


300 214 93.7353215 97.1219

302 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 4137 23 19 49 27

216 82.3724304 217 84 .107

221 85.0711306 222 85.1314

223 85.1613308 Pulse z e ro ing f a i l e d ! ! !

Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 4137 23 19 49 27

310 224 73.9409225 74 .653

312 229 74.9167230 74.9581


37 23 19 49 27316 231 74.4074

232 74.8193318 233 74.9427

236 74.9886320 237 74.9911

Pulse z e ro ing f a i l e d ! ! !322 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 41

37 23 19 49 27238 68.2949

324 239 69.3932244 69.9106

326 245 69.9213Pulse z e ro ing f a i l e d ! ! !

328 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 4137 23 19 49 27

246 84.0321330 247 86.0858

248 88.8257332 253 90.9588

254 90.9858334 255 91.0307

Pulse z e ro ing f a i l e d ! ! !336 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 41

37 23 19 49 27256 94.0572

338 257 96.8046258 97.8087

340 Zeroed Pulse I n d i c e s : 1 25 7 45 51 3 17 31 29 4137 23 19 49 27 15

259 88.8588342 260 89.5749


266 90.9354344 267 90.9555

Pulse z e ro ing f a i l e d ! ! !346 **** REFINING THE SOLUTION ****

267 98.1254348 268 98.2823

269 98.3255350 270 98.4202

271 98.5255352 Zeroed p u l s e s are : 1 2 3 4 7 8 15 16 17 18 19

20 23 24 25 26 27 28 29 30 31 32 37 38 41 4245 46 49 50 51 52

Fina l F i d e l i t y i s : 98 .5255354 Fina l GAX i s : 0 0 0 0

1 .4911 2 .6772 0 0 0.979671 .6372 4 .6514 4 .4006 3 .1373 1 .2645

0 0 0 0 00 5 .332 0.61489 0 00 0 0 0 00 0 0 1 .0635 2 .285

6 .2102 1 .3274 0 0 1 .04280.12714 0 0 3 .1448 1 .6622

0 0 4 .6146 5 .3583 00 0 0 1 .4973 0 .85770 2 .976 0 2 .065 00 0 0 0 0

3 .1412 0 0 0 01 .931 0 1 .5619 0 0

0 0 2 .9985 0 00 0

356 Elapsed time i s 708.051272 seconds .


Figure B.1: Controlled SWAP pulse sequence.

The output shown here can read using the representation scheme used for encoding

the problem solutions in GA optimization. The pulse sequence matrix constructed

from the array ‘gax’ (as given in matlab code ‘uodFitness.m’) is shown in Eqn.

B.2,

0 0 0 0 85.43 153.39 0 170.51 0

0 0 56.13 93.80 266.50 252.14 118.32 0 0

179.75 72.45 0 0 0 0 0 0 0

0 0 305.50 35.23 0 0 0 179.98 0

0 0 0 0 0 0 0 0 0

0 0 60.93 130.92 355.82 76.05 110.64 0 89.49

0 0 59.75 7.28 0 0 0 0 0

180.18 95.24 0 0 264.40 307.01 0 171.80 0

0 0 0 0 85.79 49.14 0 0 0

(B.2)

Here each row represents a pulse-delay segment with column values specifies it

parameters values. First six columns represents pulse parameters as shown in

the matlab code ‘uodFitness.m’) and last three columns represents the coupling

evolution of 1-2 coupling, 2-3 coupling and 1-3 coupling. The equivalent pulse

sequence can be shown as,

85.43 (153.39)3 - U23(170.51) - 56.13(93.80)2 : 266.50(252.14)3 - U12(118.32)

- 179.75(72.45)1 : 305.50 (35.23)2 - U23(179.98) - 60.93(130.92)2 : 355.82(76.05)3

- U12(110.64) - U13(89.49) - 59.75(7.28)2 : 180.18(95.24)1 : 264.40(307.01)3

- U23(171.80) - 85.79(49.14)3.

Graphical representation of the this pulse sequence is shown in Fig. B.1.

Appendix C

Non-Destructive Discrimination

of Arbitrary Set of Orthogonal

Quantum States by NMR using

Quantum Phase Estimation∗

C.1 Introduction

There are several theoretical protocols available for orthogonal state discrimina-

tion [134, 135]. Walgate et al. showed that, using local operations and classical

communication (LOCC) multipartite orthogonal states can be distinguished per-

fectly [134]. However if only a single copy is provided and only LOCC is allowed,

it cannot discriminate quantum states either deterministically or probabilistically

[136]. Estimation of the phase plays an important role in quantum information

processing and is a key subroutine of many quantum algorithms. When the phase

estimation is combined with other quantum algorithms, it can be employed to

perform certain computational tasks such as quantum counting, order finding and

factorization [137, 138]. Phase Estimation Algorithm has also been utilised in re-

cent important application in which the ground state of the Hydrogen molecule has

been obtained upto 45 bit accuracy in NMR and upto 20 bit accuracy in photonic

systems [63].

∗Published in AIP conference proceedings, 1384 229(2011) (Article number 1 in page vii).First work performed without GA optimization.

139

Appendix C. Non-Destructive Discrimination using Phase Estimation 140

By defining an operator with preferred eigen-values, phase estimation can be used

logically for discrimination of quantum states with certainty [3]. It preserves the

state since local operations on ancilla qubit measurements do not affect the state.

Here we describe an algorithm for non-destructive state discrimination using only

phase estimation alone. The algorithm described here is scalable and can be

used for discriminating any set of orthogonal states (entangled or non-entangled).

Earlier non-destructive Bell state discrimination has been described by Gupta et

al. [139] and verified experimentally in our laboratory by Jharana et al. [52]. Bell

states are specific example of orthogonal entangled states. The circuit used for

Bell state discrimination [139] is based on parity and phase estimation and will

not be able to discriminate a superposition state which has no definite parity. For

example consider a state |ψ〉 = 1√2(|00〉+|01〉), which belongs to a set of orthogonal

states. Here |00〉 has parity 0 and |01〉 has parity 1. Hence the above |ψ〉 does

not have a definite parity and cannot be distinguished from its other members of

the set, by the method of Gupta et al. [139]. Sec. C.2 describes the design of a

circuit for non-destructive state discrimination using phase estimation. Sec. C.2

also contains non-destructive discrimination of special cases such as Bell states and

three qubit GHZ states using phase estimation. Sec. C.3 describes experimental

implementation of the algorithm for two qubit states by NMR quantum computer

and Sec. C.4 describes the Matlab® simulation of non-destructive discrimination

of three qubit GHZ states.

C.2 Theory

For a given eigen-vector |φ〉 of a unitary operator U , phase estimation circuit

with Controlled-U operator can be used for finding the eigen-value of |φ〉 [3].

Conversely the reverse of the algorithm, with defined eigen-values can be used for

discriminating eigen-vectors. By logically defining the operators with preferred

eigen-values, the discrimination, as shown here, can be done with certainty.

C.2.1 The General Procedure (n-qubit case):

For n qubit case the Hilbert space dimension is 2n, having 2n independent orthog-

onal states. Hence we need to design a quantum circuit for state discrimination for

a set of 2n orthogonal quantum states. Consider a set of 2n orthogonal states φi,


Figure C.1: The general circuit for non-destructive Quantum State Discrimi-nation. For discriminating n qubit states it uses n number of ancilla qubits withn controlled operations. n ancilla qubits are first prepared in the state |00...0〉.Here H represents Hadamard transform and the meter represents a measure-ment of the qubit state. The original state encoded in n qubits is preserved(notdestroyed).

where i = 1, 2, ....2n. The main aim of the discrimination circuit is to make direct

correlation between the elements of φi and possible product states of ancilla

qubits. As there are 2n states, we need n ancilla qubits for proper discrimination.

The discrimination circuit requires n Controlled Operations. Selecting these n

operators Uj (where j = 1, 2, ...n) is the main task in designing the algorithm.

The set Uj depends on the 2n orthogonal states in such a way that the set of

orthogonal vectors forms the eigen-vector set of the operators, with eigen-values

±1. The sequence of +1 and −1 in eigen-values should be defined in a special way,

as outlined below. Let eij (with i = 1, 2...2n) be the eigen-value array of Uj, and

it should satisfy following conditions.

Condition #1 : Eigen-value arrays eij of all operators Uj should contain equal

number of +1 and -1,

Condition #2 : For the first operator U1, the eigen-value array ei1 can be any

possible sequence of +1 and -1 with Condition #1,

Condition #3 : The restriction on eigen-value arrays starts from Uj=2 onwards.

The eigen-value array (ei2) of operator U2 should not be equal to ei1 or its

complement, while still satisfying the Condition #1.

Condition #4 : By generalizing the Condition #3, the eigen-value array (eik)of operator Uk should not be equal to eim (m = 1, 2, ...k−1) or its complement.

manu

manu

manu

manu

manu


Let Mj be the diagonal matrix formed by eigen-value array eij of Uj. The

operator Uj is directly related to Mj by a unitary transformation given by,

Uj = V −1 ×Mj × V, (C.1)

where V is the matrix formed by the column vectors |φi〉, V = [ |φ1〉 |φ2〉|φ3〉 ..... |φn〉].

The circuit diagram for implementation of Phase Estimation Algorithm (PEA) to

discriminate orthogonal states using the Controlled-Uj operations such that the

original state is preserved for further use in any quantum circuit is shown in Fig.

C.1.

As the eigen-values defined are either +1 or −1, the final ancilla qubit states will

be in product state (without superposition), and hence can be measured with

certainty. It can be shown that the selection of specific operator set Uj with the

conditions discussed above makes direct correlation between 2n product states of

ancilla qubit and elements of |φi〉 so that ancilla measurements can discriminate

the state.

C.2.2 Single qubit case:

For a single qubit system, the Hilbert space dimension is 2. So we can discriminate

a state from a set of two orthogonal states. Consider an illustrative example with

the orthonormal set as |φ1〉 = 1√2(|0〉+ |1〉), |φ2〉 = 1√

2(|0〉−|1〉) . The quantum

circuit for this particular case can be designed by following the general procedure

discussed in Sec. C.2. The V matrix for the given states |φ1〉, |φ2〉 is,

V =1√2

(1 1

1 −1

)


According to the rules given in Sec. C.2, M can be either

(1 0

0 −1

)or

(−1 0

0 1

).

For M =

(1 0

0 −1

),

U = V −1 ×M × V =

(0 1

1 0

). (C.2)

The circuit diagram for this case is identical to Fig. C.1, having only one work and

one ancilla qubit. It can be easily shown that, the ancilla qubit measurements are

directly correlated with the input states. For the selected M1, if the given state is

|φ1〉 then ancilla will be in the state |0〉 and if the given state is |φ2〉 ancilla will

be in the state |1〉.

For a general set |φ1〉 = (α|0〉+ β|1〉), |φ2〉 = (β|0〉 − α|1〉) (where α and β are

real numbers satisfying, |α|2 + |β|2 = 1), operator U for eigenvalue array 1,−1can be shown as,

U =

(Cos(θ) Sin(θ)

Sin(θ) −Cos(θ)

), (C.3)

where θ = 2× Tan−1(βα

).

C.2.3 Two qubit case:

The Hilbert space dimension of two qubit system of is four. Consider an illustrative

example with a set of orthonormal states

|S(α, β)〉 = (α|00〉+ β|01〉), (α|10〉+ β|11〉), (β|10〉 − α|11〉), (β|00〉 − α|01〉)(C.4)

where α and β are real numbers satisfying, |α|2 + |β|2 = 1. This set is so chosen

that the states are (a)orthogonal, (b)not entangled, (c)different from Bell states,

(d)do not have definite parity and (e)contain single-superposed-qubits (SSQB)

(in this case second qubit is superposed). Using the general procedure discussed


states Ancilla-1 Ancilla-2|φ1〉 |0〉 |0〉|φ2〉 |0〉 |1〉|φ3〉 |1〉 |0〉|φ4〉 |1〉 |1〉

Table C.1: State of ancilla qubits for different input states for two qubitorthogonal states.

above, we can select the eigen-value arrays for two operators U1 and U2 as

e1 = 1, 1,−1,−1, e2 = 1,−1, 1,−1. (C.5)

U1 and U2, the unitary transformation of the diagonal matrices formed by e1and e2 are,

U1 =

Cos(θ) Sin(θ) 0 0

Sin(θ) −Cos(θ) 0 0

0 0 Cos(θ) Sin(θ)

0 0 Sin(θ) −Cos(θ)

, (C.6)

U2 =

Cos(θ) Sin(θ) 0 0

Sin(θ) −Cos(θ) 0 0

0 0 −Cos(θ) −Sin(θ)

0 0 −Sin(θ) Cos(θ)

, (C.7)

where, θ = 2× Tan−1(βα

).

The output state of the ancilla qubit run through all possible product states as

input state changes, as listed in Tab. C.1. The quantum circuit for two qubit

state discrimination is shown in Fig. C.2a.


C.2.3.1 Special case (α = β = 1√2):

The set of orthogonal states are,

|S(1√2,

1√2

)〉 = |φi〉 = 1√2

(|00〉+ |01〉), 1√2

(|10〉+ |11〉),

1√2

(|10〉 − |11〉), 1√2

(|00〉 − |01〉).(C.8)

The operators U1 and U2 can be found by substituting the value of θ = π2

in (5)

and (6),

U1 =1√2

0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0

and U2 =1√2

0 1 0 0

1 0 0 0

0 0 0 −1

0 0 −1 0

. (C.9)

The quantum circuit for the set (Eqn. C.8) is same as the general case of any set

of two qubit orthogonal states(Fig. C.2a). Experimental implementation of this

case has been performed using NMR and is described in Sec. C.3

C.2.4 Bell state discrimination:

Bell states are maximally entangled two qubit states (also known as Einstein-

Podolsky-Rosen states) [48]. They play a crucial role in several applications of

quantum computation and quantum information theory. They have been used for

teleportation, dense coding and entanglement swapping [49–51, 140]. Bell states

have also found application in remote state preparation, where a known state is

prepared in a distant laboratory [141]. Hence, it is of general interest to distinguish

Bell states without disturbing them.The complete set of Bell states are,

|Bi〉 =

1√2(|0201〉+ |1211〉), 1√

2(|0201〉 − |1211〉),

1√2(|0211〉+ |1201〉), 1√

2(|0211〉 − |1201〉)

. (C.10)

Bell states form an orthogonal set. Hence one can design a circuit for Bell

state discrimination using only phase estimation. The circuit diagram is same

as that shown in Fig. C.2a with different U1 and U2. For eigen-value arrays


e1 = 1,−1, 1,−1, e2 = −1, 1, 1,−1. U1 and U2 are obtained as,

U1 =

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

and U2 =

0 0 0 −1

0 0 1 0

0 1 0 0

−1 0 0 0

. (C.11)

The controlled operators (C−U1 and C−U2) for phase estimation which involves

3-qubit operators can be written as the product of 2-qubit operators as,

C − U1 = C −NOT 31 × C −NOT 3

2 ,

C − U2 = C − π21 × C −NOT 4

1 × C −NOT 42 × C − π2

1.(C.12)

Here qubits 1 and 2 are work qubits in which the Bell states are encoded and 3

and 4 are the ancilla qubits. Here C − NOT ij represents C-NOT operation with

control on ith qubit and target on jth qubit. The splitting of three qubit operator

into two qubit operators is needed for the implemention of C − U1 and C − U2.

There already exists an algorithm for non-destructive discrimination of Bell state

by Gupta et al. [139], which has also been experimentally implemented in NMR by

Jharana et al. [52]. The circuit of Gupta et al. [139] is based on parity and phase

measurement and will fail for a superposed state which has no definite parity.

However for non-destructive discrimination of Bell states using the present phase

estimation algorithm is similiar to Gupta’s circuit where the parity estimation is

replaced by modified phase estimation.

C.2.5 GHZ state discrimination:

GHZ states are maximally entangled multi qubit states [142]. GHZ states have

been used in several quantum algorithms such as quantum secret sharing, con-

trolled dense coding and quantum key distribution [143, 144]. These algorithms

make use of entanglement and hence it is important to discriminate GHZ states

by preserving their entanglement.


All n qubit GHZ states form an orthogonal set (without definite parity). Hence a

circuit can be designed for discriminating general n qubit GHZ states using only

phase estimation.

Consider the case of three qubit GHZ states, which are

|Gi〉 =

1√2(|000〉+ |111〉), 1√

2(|000〉 − |111〉),

1√2(|001〉+ |110〉), 1√

2(|001〉 − |110〉),

1√2(|010〉+ |101〉), 1√

2(|010〉 − |101〉),

1√2(|011〉+ |100〉), 1√

2(|011〉 − |100〉).

. (C.13)

Here we need three ancilla qubits and have to implement three controlled operators

for state discrimination. Verification of the NMR experiment to discriminate such

states has been carried out here using Matlab® and the parameters of a four qubit

NMR system, as described in Sec. C.4.

C.3 Experimental Implementation by NMR

C.3.1 Non-Destructive Discrimination of two qubit orthog-

onal states:

Experimental implementation of the quantum state discrimination(QSD) algo-

rithm has been performed here for 2 qubit case for the orthogonal set (Eqn. C.8).

The discrimination circuit diagram shown in Fig. C.2a, needs a 4 qubit system.

As two ancilla qubits are independent of each other, following [52] one can split the

experiment into 2 measurements with a single ancilla qubit (Fig.C.2b and C.2c).

The NMR implementation of the discrimination algorithm starts with (i) prepara-

tion of the pseudo-pure state followed by (ii) creation of input state (iii) Quantum

Phase estimation with operators Uj. Finally the measurement on ancilla qubits

yields the result.

The experiment has been carried out at 300K in 11.7T field in a Bruker AV 500

spectrometer using a triple resonance QXI probe. The system chosen for the

implementation of the discrimination algorithm is Carbon-13 labeled Dibromo-

fluoro methane 13CHFBr2, where 1H, 19F and 13C act as the three qubits[78].


Figure C.2: (a) Two qubit State discrimination circuit for Experimental im-plementation in three qubit NMR quantum computer, (b) and (c) are splittingof the circuit-(a) into two circuits with single ancilla measurements.

The 1H, 19F and 13C resonance frequencies at this field are 500, 470 and 125

MHz, respectively. The scalar couplings between the spins are: JHC = 224.5Hz,

JHF = 49.7Hz and JFC = −310.9Hz(Fig. C.3).

The NMR Hamiltonian for a three qubit weakly coupled spin system is [14],

H =3∑i=0

νiIiz +

3∑i<j=1

JijIizIjz , (C.14)

where νi are the Larmor frequencies and the Jij are the scalar couplings. The

starting point of any algorithm in an NMR quantum information processor is the

equilibrium density matrix, which under high temperature and high field approx-

imation is in a highly mixed state represented by[78],

ρeq ∝ γHIHz + γCI

Cz + γF I

Fz = γH(IHz + 0.94IFz + 0.25ICz ). (C.15)

There are several methods for creating pseudo pure states (PPS) in NMR from

equlibrium state [28–30, 145]. We have utilized the spatial averaging technique

[145] for creating pseudo pure states as described in [78]. The spectra for equlib-

rium and |000〉 PPS are shown in Fig. C.3.

For Phase Estimation algorithm, due to its high sensitivity, proton spin has been

utilized as the ancilla qubit; and the two qubit states, to be discriminated, are


FLUORINECARBONPROTON

a.

b.

Figure C.3: The three qubit NMR sample used for experimental implemen-tation. The nuclear spins 1H, 19F and 13C are used as the three qubits. (a)Equlibrium Spectra of proton, carbon and fluorine, (b) Spectra corresponds tothe created |000〉 pseudo pure state. These spectra are obtain by using 90o

measuring pulse on each spin.

encoded in carbon and fluorine spins. As the measurements are performed only

on ancilla qubit, we record only proton spectra for non-destructive discrimination

of the state of carbon and fluorine. The state of the ancilla qubit can be identified

by the relative phase of the spectra. We set the phase such that a positive peak

indicates that the proton was initially in state |0〉.

C.3.1.1 Implementation of Controlled-U1 and U2:

For the set of orthogonal states given in Eqn. C.8 the U1 and U2 are given in

Eqn. C.9. Let H1 and H2 be the effective Hamiltonians for Controlled-U1 and

Controlled-U2 propagators such that,

Controlled-U1 = exp(iH1),

Controlled-U2 = exp(iH2).(C.16)

where H1 and H2, in terms of product operators [14] are obtained as,

H1 = (π

4I − π

2I1z −

π

2I3x + πI1

z I3x),

H2 = (π

4I − π

2I1z − πI2

z I3x + 2πI1

z I2z I

3x)

(C.17)

Since the various terms in H1 and H2 commute with each other, one can write,


Figure C.4: The pulse sequence for Controlled-U1 and Controlled-U2 oper-ators for two qubit orthogonal states shown in Eqn. C.9 (Here narrow pulsesindicate (π2 ) pulses and broad pulses indicate π pulses with the phase givenabove the pulse).

200 0 0 0 0 200 200 200 200 200 200 200 Hz

Figure C.5: Ancilla (proton spin) spectra of final state for two qubit statediscrimination algorithm for (i) |φ1〉, (ii) |φ2〉, (iii) |φ3〉, (iv) |φ4〉 states. A1

and A2 are results of two measurements on single ancilla (Fig. C.2b and C.2crespectively) qubit-1 and 2 (here it is two experiments with same ancilla qubit).These spectra are obtained with a 90o measuring pulse on the ancilla(proton)qubit at the end of the pulse sequence.

Controlled− U1 = exp(i(π

4I − π

2I1z −

π

2I3x + πI1

z I3x)) =

exp(iπ

4I)× exp(−iπ

2I1z )× exp(−iπ

2I3x)× exp(iπI1

z I3x),

Controlled− U2 = exp(i(π

4I − π

2I1z − πI2

z I3x + 2πI1

z I2z I

3x)) =

exp(iπ

4I)× exp(−iπ

2I1z )× exp(−iπI2

z I3x)× exp(i2πI1

z I2z I

3x).

(C.18)

As the decomposed terms commute with each other, these propagators can be

easily implemented in NMR(Fig. C.4). Single spin operators such as Ix, Iy are

implemented using R.F pulses. The Iz operator is implemented using composite z

rotation pulses in NMR ((π2)−x(

π2)y(

π2)x) [95, 146]. Two spin product terms such

as I izIjx are implemented using scalar coupling Hamiltonian evolution sandwiched

between two (π2)y pulse on j spin [78]. The three spin product operator terms are

implemented using cascades of two spin operator evolutions(Tseng et al. [147]).


The experimental results are shown in Fig. C.5. Proton spectra shows the state

of ancilla qubit, which in-turn can be used for discrimination of two qubit state

in carbon and fluorine spins. Positive peaks in Fig. C.5 means ancilla is in qubit

state |0〉 and negative peak indicates ancilla qubit is in state |1〉. Thus spectra in

Fig. C.5 indicates that (i), (ii), (iii), (iv) are respectively |φ1〉, |φ2〉, |φ3〉, |φ4〉 (Tab.

C.1). To compute fidelity of the experiment, complete density matrix tomography

has been carried out (Fig. C.6). The experimental results are in agreement with

the Tab. C.1 with an ‘average absolute deviation’ [32] of 4.0% and ‘maximum

absolute deviation’ [32] of 7.2%, providing the desired discrimination.

C.4 Three qubit GHZ state Discrimination using

Matlab® Simulation:

Non-destructive discrimination of the three qubit maximally entangled (GHZ)

states using only Phase Estimation algorithm as described in Sec. C.2 in NMR

has also been performed using a Matlab® simulation. This simulation verifies

the principle involved but does not include any decoherence or pulse imperfection

effects. The three qubit GHZ states form a set Gi given by Eqn. C.13 can be

re-expressed as,

|Gi〉 = |φ1〉, |φ1〉, |φ3〉, ........|φ8〉 (C.19)

The discrimination of a 3-qubit GHZ state using phase estimation requires 3

work qubits and 3 ancilla. We divide the 6 qubit quantum circuit into three

circuits. Each circuit has three work qubits and a single ancilla. There are several

possibilities for eigen-value sets which will satisfy the sets of conditions discussed

in Sec. C.2. Consider one such set,

e1e2e3

=

e1 = 1,−1, 1,−1, 1,−1, 1,−1,e2 = −1, 1, 1,−1, 1,−1,−1, 1,e3 = −1, 1, 1,−1,−1, 1, 1,−1.

. (C.20)

For this eigen-value set (18), the Controlled− Uj operators can be written as


Figure C.6: Density Matrix Tomography of the initial and final states ofquantum state discrimination circuit. First qubit is the ancilla. It is evidentthat the state of 2nd and 3rd qubits are preserved. (here 1→ |000〉, 2→ |001〉,3→ |010〉, 4→ |011〉, 5→ |100〉, 6→ |101〉, 7→ |110〉, 8→ |111〉.)


Figure C.7: Pulse sequence for Controlled operators in GHZ state discrim-ination. (Here narrow pulses indicate (π2 ) pulses and broad pulses indicate πpulses with the phase given above the pulse).


Figure C.8: The chemical structure, the chemical shifts and spin-spin couplingof a 13C labelled Crotonic Acid. The four 13C spins act as four qubits [2].

state Measurement-1 Measurement-2 Measurement-3|φ1〉 |0〉 |1〉 |1〉|φ2〉 |1〉 |0〉 |0〉|φ3〉 |0〉 |0〉 |0〉|φ4〉 |1〉 |1〉 |1〉|φ5〉 |0〉 |0〉 |1〉|φ6〉 |1〉 |1〉 |0〉|φ7〉 |0〉 |1〉 |0〉|φ8〉 |1〉 |0〉 |1〉

Table C.2: State of ancilla qubits for different input states of Eqn. C.13 andC.19.

Controlled− U1 = C −NOT a1 × C −NOT a2 × C −NOT a3 ,

Controlled− U2 = C − π23 × C −NOT a1 × C −NOT a2×

C −NOT a3 × C − π23,

Controlled− U3 = C − π13 × C −NOT a1 × C −NOT a2×

C −NOT a3 × C − π13.

(C.21)

Here 1,2 and 3 are the work qubits, in which the GHZ state is encoded and ‘a’ is

the ancilla qubit. Pulse sequence for these operators are shown in Fig. C.7. The

results of ancilla qubit measurements are tabulated in Tab. C.2.

NMR simulation has been carried out using the parameters of a well known 4-qubit

system, crotonic acid with all carbons labelled by 13C (Fig. C.8) [2]. The density

matrix tomography of the Matlab® experiment for a few selected(|φ1〉, |φ4〉 and

|φ7〉) GHZ states are shown in Fig. C.9. This confirms that the method of Phase

Estimation discussed in Sec. C.2 can be used for discrimination of GHZ states

without destroying them.


First Experiment Second Experiment Third Experiment

(i) |0〉a( 1√2(|000〉+ |111〉)) |1〉a( 1√

2(|000〉+ |111〉)) |1〉a( 1√

2(|000〉+ |111〉))

2 4 6 8 10 12 14 16

24

68

1012

1416

−0.5

−0.25

0

0.25

0.5

2 4 6 8 10 12 14 16

24

68

1012

1416

−0.5

−0.25

0

0.25

0.5

2 4 6 8 10 12 14 16

24

68

1012

1416

−0.5

−0.25

0

0.25

0.5

(ii) |1〉a( 1√2(|001〉 − |110〉)) |1〉a( 1√

2(|001〉 − |110〉)) |1〉a( 1√

2(|001〉 − |110〉))

2 4 6 8 10 12 14 16

24

68

1012

1416

−0.5

−0.25

0

0.25

0.5

2 4 6 8 10 12 14 16

24

68

1012

1416

−0.5

−0.25

0

0.25

0.5

2 4 6 8 10 12 14 16

24

68

1012

1416

−0.5

−0.25

0

0.25

0.5

Figure C.9: Matlab® simulation results for GHZ state discrimination. Thesimulated spectras are shown for three GHZ states |φ1〉 and |φ4〉. It is evidentfrom final density matrix that the GHZ states are preserved. (Here 1→ |0000〉,2 → |0001〉, 3 → |0010〉, 4 → |0011〉, 5 → |0100〉, 6 → |0101〉, 7 → |0110〉,8 → |0111〉, 9 → |1000〉, 10 → |1001〉, 11 → |1010〉, 12 → |1011〉, 13 → |1100〉,14→ |1101〉, 15→ |1110〉, 16→ |1111〉. First qubit is the ancilla)

C.5 Conclusion

A general scalable method for non-destructive quantum state discrimination of

a set of orthogonal states using quantum phase estimation algorithm has been

descibed, and experimently implemented for a two qubit case by NMR. As the

direct measurements are performed only on the ancilla, the discriminated states

are preserved. The generalization of the algorithm is illustrated by discrimination

of GHZ states using a Matlab® simulation.

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Index

adaptive feasible, 20

adiabatic evolution, 41

adiabatic quantum computation, 9

AED, 52, 53

angular momentum, 2

APT, 89

average experimental deviation, 50

average Hamiltonian, 50, 113

Avogadro’s number, xxvii

BB-INEPT, 90

Bell state, 32, 140, 143, 145

Bloch, 1

Bloch sphere, 7, 8

Bodenhausen, 90

Bohr magneton, xxvii

Boltzmann constant, xxvii

Boltzmann distribution, 11

Boltzmann statistics, 6

chemical shift, 4, 27, 91

coherence transfer, 69

composite pulse, 69, 72, 92

concurrence, 53

controlled controlled NOT, 104, 105

controlled Hadamard, 103, 104

controlled NOT, 9, 28, 30, 31, 69, 72, 86

controlled operators, 9, 141, 153

controlled phase, 41, 49, 51

controlled SWAP, 105, 106, 113, 138

cost function, 14

crossover, 16, 19, 92, 116

crossover fraction, 92

crossover heuristic, 19

crossover intermediate, 19

crossover scattered, 19

D WAVE, 10

Darwinian evolution, 14

David. J. Wineland, 7

decision variable, 14

decoherence, 10, 53

delay zeroing, 116

dense coding, 145, 146

density matrix, 11, 12

density matrix tomography, 151, 152

DEPT, 96, 99

Deutsch-Jozsa algorithm, 10, 38

diamagnetic shielding, 3, 4

dipolar coupling, 5

Dzyaloshinskii-Moriya, 43

Einstein-Podolsky-Rosen, 145

electron mass, xxvii

elementary charge, xxvii

elite count, 92, 115

elite individuals, 16

entanglement dynamics, 53

entanglement preservation, 54, 55

entanglement swapping, 145

equilibrium state, 11

Ernst, 1

excitation, 72

expm, 71

171

Index 172

factorization, 38

ferromagnetism, 43

Feynman, 37

fidelity profile optimization, 37–39, 42,

44, 51, 52

fidelity space, 38

fitness function, 16, 25, 26, 39, 72, 73,

92, 93

GAQIC, 20, 90, 91, 93, 98

genetic algorithm, 16

GHZ state, 140, 146, 151

gradient ascent pulse engineering, 23

Grover’s search algorithm, 11

Hadamard gate, 32

Heisenberg-XY, 39, 46, 49, 52

Hilbert space, 140, 143

homonuclear NMR, 27

INEPT, 89, 91

inversion, 72

Jharana, 146

John Holland, 14

Leggett-Garg inequality, 11, 38

linear spin chain, 52

LOCC, 139

logical labelling, 12

long lived singlet state, 24

longitudinal relaxation time, 49

magnetic dipoles, 5

Mathematica, 70

Matlab, 74, 83

methine, 91

methyl, 91

methylene, 91

min-max difference, 93, 96

MRI, 1

mutation, 16, 92, 115

mutation gaussian, 20

mutation operators, 19

no-hiding theorem, 11

NOE, 89

non-destructive discrimination, 146

NOON, 57

Nyberg, 90

objective function, 14, 16

original modulation, 96

parity estimation, 146

Pauli, 1

Pauli matrices, 8

permeability, xxvii

phase estimation algorithm, 139, 140,

142, 145–148

Planck constant, xxvii

polarization transfer, 72, 89–91

population inversion, 69

proton mass, xxvii

pseudo pure state, 11, 30, 32, 50, 147–

149

pulse zeroing, 116

pulsed field gradients, 24

Purcell, 1

Q-INEPT-CT, 90, 100

QIP, 37

quadrupolar interaction, 5

quantification, 89, 91

quantification efficiency, 93, 95, 96

quantitative INEPT, 16, 20

quantum counter, 106–108

Index 173

quantum gates, 8

quantum key distribution, 146

quantum probability splitter, 110, 111

quantum secret sharing, 146

quantum simulation, 11

quantum state discrimination, 140, 141

quantum state tomography, 12, 34, 50

Rabi, 1

recombination operators, 14

refocused INEPT, 91, 95–97

remote state preparation, 145

representation scheme, 24, 72, 92

reproduction operators, 18

rf pulse, 5

Roulette, 17, 116

SALLT, 12

scalar coupling, 4

Serge Haroche, 7

Shor’s algorithm, 11

single point crossover, 19

single qubit rotation, 8, 27

singlet state, 30, 32, 34

spatial averaging, 12, 30, 50, 148

spectral editing, 97

spin quantum numbers, 2

spin selective pulses, 26

spin-lock, 34

spin-orbit coupling, 43

SSQB, 143

state to state optimization, 24, 30

Stern and Gerlach, 1

stochastic uniform, 17

stopping criteria, 16

strongly modulated pulses, 23

superconductor, 10

teleportation, 145

temporal averaging, 12

thermal equilibrium, 6, 30, 34

time restricted modulation, 96

Torres, 90

transfer functions, 91

transverse relaxation time, 49

trapped ions, 10

two point crossover, 19

UOD, 38, 103

W state, 109, 110

WALTZ-16, 34

Zeeman, 2

Zeeman interaction, 4

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Optimization of NMR Experiments using Genetic Algorithm: …spinhalf.in/bio_manu/manu_Thesis.pdf · 2019-02-04 · I thank Prof. Apoorva Patel for his introductory courses on Quantum