Anil KumarAnil Kumar11, K. Rama Koteswara Rao, K. Rama Koteswara Rao11, T.S. Mahesh, T.S. Mahesh22 and and V.S.ManuV.S.Manu11

Indian Institute of Science, Bangalore-560012Indian Institute of Science, Bangalore-560012Indian Institute of Science Education and Research, Pune-411 008Indian Institute of Science Education and Research, Pune-411 008

Quantum Simulations by NMR: Quantum Simulations by NMR: Recent DevelopmentsRecent Developments

QIPA-December 2013, HRI-Allahabad

Simulating a Molecule: Using Aspuru-Guzik Adiabatic Algorithm

(i) J.Du, et. al, Phys. Rev. letters 104, 030502 (2010).

Used a 2-qubit NMR System ( 13CHCl3 ) to calculated the ground state energy of Hydrogen Molecule up to 45 bit accuracy.

(ii) Lanyon et. al, Nature Chemistry 2, 106 (2010).

Used Photonic system to calculate the energies of the ground and a few excited states up to 20 bit precision.

Recent Developments

In 1982, Feynman suggested that it might be possible to simulate the evolution of quantum systems efficiently, using a quantum

simulator

Experimental Techniques for Quantum Computation:1. Trapped Ions

4. Quantum Dots

3. Cavity Quantum Electrodynamics

(QED)

6. NMR

Quantum Dots:Quantum Dots:

mElectrodesDot

Circle

http://theorie5.physik.unibas.ch/qcomp/node3.html

http://news.uns.purdue.edu/html4ever/010917.Chang.quantum.html

7. Josephson junction qubits

8. Fullerence based ESR quantum computer

5. Cold Atoms

2. Polarized Photons Lasers

0

1. Nuclear spins have small magnetic moments and behave as tiny quantum magnets.

2. When placed in a magnetic field (B0), spin ½ nuclei orient either along the field (|0 state) or opposite to the field (|1 state) .

4. Spins are coupled to other spins by indirect spin-spin (J) coupling, and controlled (C-NOT) operations can be performed using J-coupling.

Multi-qubit gates

Nuclear Magnetic Resonance (NMR)

3. A transverse radio-frequency field (B1) tuned at the Larmor frequency of spins can cause transition from |0 to |1 (NOT Gate by a 1800 pulse). Or put them in coherent superposition (Hadamard Gate by a 900 pulse). Single qubit gates.

NUCLEAR SPINS ARE QUBITS

B1

DSX 3007.0 Tesla

AMX 4009.4 Tesla

AV 50011.7 Tesla

AV 70016.5 Tesla

DRX 50011.7 Tesla

NMR Research Centre, IISc1 PPB

Field/ Frequency stability = 1:10 9

NMR Facility in IISc800 MHz NMR spectrometer equipped with cryogenic probe

Installed in April 2013

18.8 Tesla

Why NMR?Why NMR?

> A major requirement of a quantum computer is that the coherence should last long.

> Nuclear spins in liquids retain coherence ~ 100’s millisec

and their longitudinal state for several seconds.

> A system of N coupled spins (each spin 1/2) form an N qubit Quantum Computer.

> Unitary Transform can be applied using R.F. Pulses and J-evolution and various logical operations and quantum algorithms can be implemented.

NMR sample has ~ 1018 spins.

Do we have 1018 qubits?

No - because, all the spins can’t beindividually addressed.

Spins having different Larmor frequencies can be addressed in the frequency domain resulting-in as many “qubits” as Larmor

frequencies, each having 1018 spins. (ensemble computing)

Progress so far

One needs resolved couplings between the spins in order to encode information as qubits.

Addressability in NMR

NMR Hamiltonian

H = HZeeman + HJ-coupling

= izi Jij Ii

Ij

i i < j

Weak coupling Approximation

ijJij

Two Spin System (AM)

A2 A1 M2 M1

A M

M1= A

M2= A

A1= M

A2= M

H = izi Jij Izi Izj

Under this approximation spins having same Larmor Frequency

can be treated as one Qubit

i i < j

Spin States are eigenstates

13CHFBr2

An example of a three qubit system.

1H = 500 MHz 13C = 125 MHz 19F = 470 MHz

13C

Br (spin 3/2) is a quadrupolar nucleus, is decoupled from the rest of the spin system and can be ignored.

JCH = 225 HzJCF = -311 HzJHF = 50 Hz

NMR Qubits

1 Qubit1 Qubit

00

0110

11

0

1

CHCl3

000

001010

011

100

101110

111

2 Qubits2 Qubits 3 Qubits3 Qubits

Homo-nuclear spins having different Chemical shifts (Larmor frequencies) also form multi-qubit systems

Pseudo-Pure StatePseudo-Pure State

Pure State

Tr(ρ ) = Tr ( ρ2 ) = 1

For a diagonal density matrix, this condition requires that all energy levels except one have zero populations

Such a state is difficult to create in room temperature liquid-state NMR

We create a state in which all levels except one have EQUAL populations. Such a state mimics a pure state.

ρ = 1/N ( α1 - Δρ )

Under High Temperature Approximation

Here α = 105 and U 1 U-1 = 1

0

1

2

1

0

0

4

0

1. Preparation ofPseudo-Pure States

2. Quantum Logic Gates

3. Deutsch-Jozsa Algorithm

4. Grover’s Algorithm

5. Hogg’s algorithm

6. Berstein-Vazirani parity algorithm

7. Quantum Games

8. Creation of EPR and GHZ states

9. Entanglement transfer

Achievements of NMR - QIP

10. Quantum State Tomography

11. Geometric Phase in QC

12. Adiabatic Algorithms

13. Bell-State discrimination

14. Error correction

15. Teleportation

16. Quantum Simulation

17. Quantum Cloning

18. Shor’s Algorithm

19. No-Hiding Theorem

Maximum number of qubits achieved in our lab: 8

Also performed in our Lab.

In other labs.: 12 qubits; Negrevergne, Mahesh, Cory, Laflamme et al., Phys. Rev. Letters, 96, 170501 (2006).

Recent Publications from our Laboratory

1. Experimental Test of Quantum of No-Hiding theorem.Jharana Rani Samal, Arun K. Pati and Anil Kumar, Phys. Rev. Letters, 106, 080401 (25 Feb., 2011)

2. Use of Nearest Neighbor Heisenberg XY interaction for creation of entanglement on end qubits in a linear chain of 3-qubit system.

K. Rama Koteswara Rao and Anil Kumar IJQI, 10, 1250039 (2012).

3. Multi-Partite Quantum Correlations Revel Frustration in a Quantum Ising Spin System.

K. Rama Koteswara Rao, Hemant Katiyar, T.S. Mahesh, Aditi Sen (De), Ujjwal Sen and Anil Kumar: Phys. Rev A 88, 022312 (2013).

4. Use of Genetic Algorithm for QIP. V.S. Manu and Anil Kumar, Phys. Rev. A 86, 022324 (2012)

No-Hiding Theorem

S.L. Braunstein & A.K. Pati, Phys.Rev.Lett. 98, 080502 (2007).

Any physical process that bleaches out the original information is called “Hiding”. If we start with a “pure state”, this bleaching process will yield a “mixed state” and hence the bleaching process is “Non-Unitary”. However, in an enlarged Hilbert space, this process can be represented as a “unitary”.

The No-Hiding Theorem demonstrates that the initial pure state, after the bleaching process, resides in the ancilla qubits from which, under local unitary operations, is completely transformed to one of the ancilla qubits.

Quantum Circuit for Test of No-Hiding Theorem using State Randomization (operator U).

H represents Hadamard Gate and dot and circle represent CNOT gates.

After randomization the state |ψ> is transferred to the second Ancilla qubit proving the No-Hiding Theorem.

(S.L. Braunstein, A.K. Pati, PRL 98, 080502 (2007).

NMR Pulse sequence for the Proof of No-Hiding Theorem

The initial State ψ is prepared for different values of θ and φ

Jharana Rani et al

Pulse Sequence for U obtained by using the method out lined in A Ajoy, RK Rao, A Kumar, and P Rungta, Phys. Rev. A 85, 030303(R) (2012)

Experimental Result for the No-Hiding Theorem. The state ψ is completely transferred from first qubit to the third qubit

325 experiments have been performed by varying θ and φ in steps of 15o

All Experiments were carried out by Jharana (Dedicated to her memory)

Input State

Output State

s

s

S = Integral of real part of the signal for each spin

Jharana Rani Samal, Arun K. Pati and Anil Kumar,Phys. Rev. Letters, 106, 080401 (25 Feb., 2011)

Use of nearest neighbor Heisenberg-XY interaction.

Solution

Use Nearest Neighbor Interactions

Until recently we had been looking for qubit systems, in which all qubits are coupled to each other with unequal couplings, so that all transitions are resolved and we have a complete access

to the full Hilbert space.

However it is clear that such systems are not scalable, since remote spins will not be coupled.

Koteswara Rao et al, IJQI, 10 (4) 1250039 (2012)

Heisenberg XY interaction is normally not present in liquid-state NMR: We have only ZZ interaction.

We create the XY interaction by transforming the ZZ interaction by the use of RF pulses.

In a first study towards this goal, we took a linear chain of 3-qubits

Entangle end qubits as well as Entangle all the 3 qubits

And Using only the Nearest-Neighbor

Heisenberg XY Interaction,

1

1112

1 )(N

i

yi

yi

xi

xiiXY JH

1

111

N

iiiiiiJ

Nearest-Neighbour Heisenberg XY Interaction

3232212121

yyxxyyxxXY JH

tiH XYetU )(

32212)( yyxxi Jt

A etU 3221

2)( xxyyi Jt

B etU

32212

32212)()()( xxyy

iyyxx

i JtJtBA eetUtUtU

0, 32213221 xxyyyyxx

σjx/y are the Pauli spin matrices and J is the coupling constant between two spins.

21 3 J J

Divide the HXY into two commuting parts

Consider a linear Chain of 3 spins with equal couplings

Jingfu Zhang et al., Physical Review A, 72, 012331(2005)

10000000

0cos)2sin(0sin000

0)2sin()2cos(0)2sin(000

000cos0)2sin(sin0

0sin)2sin(0cos000

000)2sin(0)2cos()2sin(0

000sin0)2sin(cos0

00000001

)(

2

2

2

22

2

2

2

2

2

2

22

2

2

2

i

ii

i

i

ii

i

tU

φ = Jt/√2

The operator U in Matrix form for the 3-spin system

where

Jingfu Zhang et al., Physical Review A, 72, 012331(2005)

10000000

02

1

20

2

1000

02

002

000

0002

10

22

10

02

1

20

2

1000

0002

002

0

0002

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10

00000001

i

ii

i

i

ii

i

tU 2132

11001101 iU

2132

00110010 iU

,22

J

tWhen

4/

Two and three qubit Entangling Operators

Bell States

10000000

079.03

021.0000

033

10

3000

00079.003

21.00

021.03

079.0000

0003

03

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30

00021.003

79.00

00000001

)(

i

ii

i

i

ii

i

tU

,2

)2(tan

1

JtWhen

2

)2(tan 1

110011101101)(333

1 iitU

1100111013

13

13

1

Phase Gate on 2ndspin

1100 i

W - State

Sample

Equilibrium spectra

Experiment

Pulse sequence to prepare PPS (using only the nearest neighbour couplings)

1H

13C

19F

Gz

Tomography of 010 > PPS

Pulse sequence to simulate XY Hamiltonian

1H

13C

19F

Bell state on end qubits

Attenuated Correlation(c)=0.86

Attenuated Correlation(c)=0.88

K R K Rao, and Anil Kumar, Int. J. Quant. Inf. 10, 1250039 (2012).

Experimental results

2132

11001101 iU

2132

00110010 iU

W-state

Attenuated Correlation(c)=0.89

Attenuated Correlation(c)=0.81

Experimental results

K R K Rao, and Anil Kumar, Int. J. Quant. Inf. 10, 1250039 (2012).

1100111013

13

13

1

GHZ-state ( |000> + |111> )/√2

Quantum simulation of frustrated Ising spins by NMR

K. Rama Koteswara Rao1, Hemant Katiyar3, T.S. Mahesh3, Aditi Sen (De)2, Ujjwal Sen2 and

Anil Kumar1:

Phys. Rev A 88 , 022312 (2013).

1 Indian Institute of Science, Bangalore2 Harish-Chandra Research Institute, Allahabad3 Indian Institute of Science Education and Research, Pune

A spin system is frustrated when the minimum of the system energy does

not correspond to the minimum of all local interactions. Frustration in

electronic spin systems leads to exotic materials such as spin glasses and

spin ice materials.

If J is negative Ferromagnetic

If J is positive Anti-ferromagnetic

The system is frustrated

3-spin transverse Ising system

The system is non-frustrated

This rotation was realized by a numerically optimized amplitude and phase modulated

radio frequency (RF) pulse using GRadient Ascent Pulse Engineering (GRAPE)

technique1.

1N. Khaneja and S. J. Glaser et al., J. Magn. Reson. 172, 296 (2005).

Diagonal elements are chemical shifts and off-diagonal elements are couplings.

Here, we simulate experimentally the ground state of a 3-spin system in both

the frustrated and non-frustrated regimes using NMR.

Experiments at 290 K in a 500 MHz NMR Spectrometer of IISER-Pune

Non-frustrated Frustrated

Multipartite quantum correlations

Non-frustrated regime: Higher

correlations

Frustrated regime:Lower

correlations

Entanglement Score using deviation Density matrix

Quantum Discord Score using full density matrix

Ground StateGHZ State (J >> h)

( > -000׀111׀ >)/√2

Fidelity = .984

Initial State:Equal Coherent Superposition State. Fidelity = .99

Koteswara Rao et al. Phys. Rev A 88 , 022312 (2013).

Recent Developments in our Laboratory:

1. Simulation of Mirror Inversion of Quantum States in an XY spin-chain.

2. Quantum Simulation of Dzyaloshinsky-Moriya (DM) interaction in presence of Heisenberg XY interaction for study of Entanglement Dynamics

(To be published)

Mirror Inversion of quantum states in an XY spin chain*

• Entangled states of multiple qubits can be transferred from one end of the chain to the other end

J1 J2 JN-1NN-1321

*Albanese et al., Phys. Rev. Lett. 93, 230502 (2004)*P Karbach, and J Stolze et al., Phys. Rev. A 72, 030301(R) (2005)

• The above XY spin chain generates the mirror image of any input state up to a phase difference.

NMR Hamiltonian of a weakly coupled spin system

Control Hamiltonian

Simulation

In practice

1) GRAPE algorithm

2) An algorithm by A Ajoy et al. Phys. Rev. A 85, 030303(R) (2012)

Here, we use a combination of these two algorithms to simulate the unitary evolution of the XY spin chain

Simulation

4-spin chain

5-spin chain

In the experiments, each of these decomposed operators are simulated using GRAPE technique

N-spin chainsFor odd N

For even N

The number of operators in the decomposition increases only linearly with the number of spins (N).

Molecular structure and Hamiltonian parameters

The dipolar couplings of the spin system get scaled down by the order parameter (~ 0.1) of the liquid-crystal medium.

The sample 1-bromo-2,4,5-trifluorobenzene is partially oriented in a liquid-crystal medium MBBA

The Hamiltonian of the spin system in the doubly rotating frame:

5-spin system

Experiment

Quantum State Transfer: 4-spin pseudo-pure initial states

Diagonal part of the deviation density matrices (traceless)The x-axis represents the standard computational basis in decimal form

Coherence Transfer: 5-spin initial states

Spectra of Fluorine spins Proton spins

K R K Rao, T S Mahesh, and A Kumar, (under review), Phys. Rev. A

Coherence Transfer: 5-spin initial states

Spectra of Fluorine spins Proton spins

Entanglement Transfer

Reconstructed deviation density matrices (trace less) of spins 1 and 2, and spins 4 and 5.

Initial States Final States

K R K Rao, T S Mahesh, and A Kumar, (under review), Phys. Rev. A

Entanglement Transfer

Reconstructed deviation density matrices (trace less) of spins 1 and 2, and spins 4 and 5.

Initial States Final States

The effective transverse relaxation times (T2*): 40-100ms 110-150ms

Fluorine transitions

Proton transitions

The Fidelity of the Final state is high. This means that the state is reproduced with high fidelity. But the attenuated correlations are low. This means the intensities of the signal in the final state are low compared to initial state. This is mainly due to decoherence.

The Genetic Algorithm

John HollandCharles Darwin 1866 1809-1882

“Genetic Algorithms are good at taking large, potentially huge, search spaces and navigating them, looking for optimal combinations of things, solutions one might not otherwise find in a lifetime”

Genetic Algorithm

Here we apply Genetic Algorithm to Quantum Information Processing Quantum Information Processing

We have used GA for

(1) Quantum Logic Gates (operator optimization)and

(2) Quantum State preparation (state-to-state optimization)

V.S. Manu et al. Phys. Rev. A 86, 022324 (2012)

Representation Scheme

Representation scheme is the method used for encoding the solution of the problem to individual genetic evolution. Designing a

good genetic representation is a hard problem in evolutionary computation. Defining proper representation scheme is the first

step in GA Optimization*.

In our representation scheme we have selected the gene as a combination of

(i) an array of pulses, which are applied to each channel with amplitude (θ) and phase (φ),

(ii) An arbitrary delay (d).

It can be shown that the repeated application of above gene forms the most general pulse sequence in NMR

* Whitley, Stat. Compt. 4, 65 (1994)

The Individual, which represents a valid solution can be represented as a matrix of size (n+1)x2m. Here ‘m’ is the number of genes in each individual and ‘n’ is the number of channels (or spins/qubits).

So the problem is to find an optimized matrix, in which the optimality condition is imposed by a “Fitness Function”

Fitness function

In operator optimization

GA tries to reach a preferred target Unitary Operator (Utar) from an initial random guess pulse sequence operator (Upul).

Maximizing the Fitness function

Fpul = Trace (Upul Χ Utar )

In State-to-State optimization

Fpul = Trace { U pul (ρin) Upul (-1) ρtar † }

Single Qubit Rotation in a Two-qubit Homonuclear case

H = 2π δ (I1z – 12z) + 2π J12 (I1zI2z)

δ = 500 Hz, J= 3.56 Hz

φ1 = 2π, φ2 = π, Θ = π/2, φ = π/2

φ1 = π, φ2 = 0Θ = π/2, φ = π/2

π/2 π/2

Simulated using J = 0

Hamiltonian used

Non-Selective (Hard) Pulses applied at the centre

Sequence obtained by Genetic Algorithm

By changing φ1 and φ2 one can excite either spin 1 or 2

Fidelity for finite J/δ

C-NOT

Equilibrium

00

01 1011

1

-1

00

00

01 1011

1

-10

0

00

01 1011

1

-1 0

0

00

01 1011

1

-1

0

000

01 1011

1

-1

0

0

Sequence obtained by Genetic Algorithm

Pseudo Pure State (PPS) creation

All unfilled rectangles represent 900 pulseThe filled rectangle is 1800 pulse. Phases are given on the top of each pulse.

Fidelity w.r.t. to J/δ

00 01 10 11

Sequence obtained by Genetic Algorithm

00 and 11 PPS are obtained by -/+ on 180 pulse. Other PPS are obtained by using an additional 180 SQR pulse

Bell state creation: From Equilibrium (No need of PPS)

The Singlet Bell State

Experimental Fidelity > 99.5 %

Shortest Pulse Sequence for creation of Bell States directly from Equilibrium

All blank pulses are 900 pulses. Filled pulse is a 1800 pulse.Phases and delays Optimized for best fidelity.

T1=8.7 sTs=11.2 sV.S. Manu et al. Phys. Rev. A 86, 022324 (2012)

Sequence obtained by Genetic Algorithm

(b) Quantum Simulation of Dzyaloshinsky-Moriya (DM) interaction (HDM) in presence of Heisenberg XY interaction (HXY) for study of Entanglement Dynamics

Manu et al. (Under Review)

DM Interaction1,2 Anisotropic antisymmetric exchange interaction arising from spin-orbit coupling. Proposed by Dzyaloshinski to explain the weak ferromagnetism of antiferromagnetic crystals (Fe2O3, MnCO3).

Quantum simulation of a Hamiltonian H requires unitary operator decomposition (UOD) of its evolution operator, (U = e-iHt) in terms of experimentally preferable unitaries.

Using Genetic Algorithm optimization, we numerically evaluate the most generic UOD for DM interaction in the presence of Heisenberg XY interaction.

1. I. Dzyaloshinsky, J. Phys & Chem of Solids, 4, 241 (1958).2. T. Moriya, Phys. Rev. Letters, 4, 228 (1960).

Decomposing the U in terms of Single Qubit Rotations (SQR) and ZZ- evolutions.

SQR by Hard pulse

ZZ evolutions by Delays

The Hamiltonian

Heisenberg XY interaction DM interaction

Evolution Operator:

Fidelity is defined as.

A. The Decomposition for ϒ = 0 -1:

1

Manu et al. (under Review).Fidelity > 99.99 %

Period of U(ϒ,τ)

This has a maximum value of 12.59. Optimization is performed for τ -> 0-15, which includes one complete period.

2

When ϒ > 1 -> ϒ’ < 1 ϒ’ = 1/ ϒ

Using above decomposition, we studied entanglement preservation in a two qubit system.

Hou et al. 1 demonstrated a mechanism for entanglement preservation using H(J,D). They showed that preservation of initial entanglement is performed by free evolution interrupted with a certain operator O, which makes the state to go back to its initial state.

1Hou et al. Annals of Physics, 327 292 (2012)

Entanglement Preservation

Without Operator O With Operator O

concurrence

µi are eigen values of the operator ρSρ*S, where S= σ1y ⊗ σ2y

Entanglement (concurrence) oscillates during Evolution.

Entanglement (concurrence) is preserved during Evolution. This confirms the Entanglement preservation method of Hou et al.

(Manu et al. Under Review)

Other IISc CollaboratorsOther IISc Collaborators

Prof. Apoorva PatelProf. K.V. RamanathanProf. N. Suryaprakash

Prof. Malcolm H. Levitt - UKProf. P.Panigrahi IISER KolkataProf. Arun K. Pati HRI-AllahabadProf. Aditi Sen HRI-AllahabadProf. Ujjwal Sen HRI-AllahabadMr. Ashok Ajoy BITS-Goa-MIT

Prof. ArvindProf. Kavita DoraiProf. T.S. MaheshDr. Neeraj SinhaDr. K.V.R.M.MuraliDr. Ranabir DasDr. Rangeet Bhattacharyya

- IISER Mohali- IISER Mohali- IISER Pune- CBMR Lucknow- IBM, Bangalore- NCIF/NIH USA- IISER Kolkata

Dr.Arindam Ghosh - NISER BhubaneswarDr. Avik Mitra - Philips BangaloreDr. T. Gopinath - Univ. MinnesotaDr. Pranaw Rungta - IISER MohaliDr. Tathagat Tulsi – IIT Bombay

Acknowledgements

Other CollaboratorsOther Collaborators

Funding: DST/DAE/DBT

Former QC- IISc-Associates/Students

This lecture is dedicated

to the memory of Ms. Jharana Rani Samal* (*Deceased, Nov., 12, 2009)

Current QC IISc - Students

Mr. K. R. Koteswara Rao Mr. V.S. Manu

Thanks: NMR Research Centres at IISc, Bangaloreand IISER-Pune for spectrometer time

Thank YouThank You

/2 ,2/When Jt

Quantum State Transfer

Two qubit Entangling Operator

,22

J

tWhen

4/

Three qubit Entangling Operator

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)2(tan

1

JtWhen

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)2(tan 1