SchrSchrödinger’s ödinger’s Elephants & Quantum Elephants & Quantum

Slide RulesSlide Rules

A.M. Zagoskin A.M. Zagoskin (FRS RIKEN & UBC)(FRS RIKEN & UBC)

S. Savel’evS. Savel’ev (FRS RIKEN & Loughborough U.)(FRS RIKEN & Loughborough U.) F. NoriF. Nori (FRS RIKEN & U. of Michigan)(FRS RIKEN & U. of Michigan)

Solving NP-complete problems with Solving NP-complete problems with approximate adiabatic evolutionapproximate adiabatic evolution

Standard quantum Standard quantum computationcomputation

Consecutive application of Consecutive application of unitary transformations unitary transformations (quantum gates)(quantum gates)

Problem Problem encodedencoded in the in the initial state of the systeminitial state of the system

Solution Solution encodedencoded in the in the final state of the systemfinal state of the system

digitaldigital operation operation

Examples: Examples: Shor’s algorithmShor’s algorithmGrover’s algorithmGrover’s algorithm

quantum Fourier transformquantum Fourier transform

Precise time-domain Precise time-domain manipulations manipulations complex design and complex design and extra sources of extra sources of decoherencesdecoherences

Problem and solution encoded in Problem and solution encoded in fragile strongly entangled states fragile strongly entangled states of the systemof the system effective decoherence effective decoherence time time must be largemust be large

Quantum error-correction (to Quantum error-correction (to extend the coherence time of the extend the coherence time of the system)system) overhead (threshold overhead (threshold theorems: 10theorems: 1044-10-101010(!)) (!))

Aharonov, Kitaev & Aharonov, Kitaev & Preskill, Preskill, quant-ph/05102310quant-ph/05102310

Adiabatic quantum Adiabatic quantum computationcomputation

ContinuousContinuous adiabatic adiabatic evolution of the systemevolution of the system

Problem Problem encodedencoded in the in the HamiltonianHamiltonian of the system of the system

Solution Solution encodedencoded in the final in the final ground stateground state of the system of the system

Farhi et al., quant-Farhi et al., quant-ph/0001106; Science ph/0001106; Science 292292(2001)472(2001)472

The approach is equivalent to The approach is equivalent to the standard quantum the standard quantum computingcomputing

Aharonov et al., Aharonov et al., quant-quant- ph/0405098ph/0405098

““Space-time swap”: the time-Space-time swap”: the time-domain structure of the domain structure of the algorithm is translated to the algorithm is translated to the time-independent structural time-independent structural properties of the systemproperties of the system

Ground state is relatively robustGround state is relatively robust much easier conditions much easier conditions on the system and its evolutionon the system and its evolution

Well suited for the realization by Well suited for the realization by superconducting quantum superconducting quantum circuitscircuits

Kaminsky, Lloyd & Kaminsky, Lloyd & Orlando,Orlando,quant-ph/0403090quant-ph/0403090Grajcar, Izmalkov & Il’ichev, Grajcar, Izmalkov & Il’ichev, PRB PRB 7171(2005)144501(2005)144501

Travelling salesman’s Travelling salesman’s problem*problem*

NN points with distances points with distances ddijij

Let Let nniaia=1=1 if if ii is stop # is stop #aa and and 00 otherwise; there are otherwise; there are NN22 variables variables nniaia ((i,a = 1,…,Ni,a = 1,…,N))

The total length of the tourThe total length of the tour aij

ajajiaij nnndL,

1,1, )(2

1

i

iaa

ia nn 1 and 1

*See e.g. M. Kastner, Proc. IEEE 93 (2005) 1765

Travelling salesman’s Travelling salesman’s problemproblem

The cost functionThe cost function

22

,1,1,

112

)(2

1

i aia

a iia

aijajajiaijts

nn

nnndH

Travelling salesman’s Travelling salesman’s problemproblem

Ising HamiltonianIsing Hamiltonian

22

,

1,1,21

21

21

2

)1)((2

1

i a

iaz

a i

iaz

aij

ajz

ajz

iaz

ijts

NN

dH

Spin HamiltonianSpin Hamiltonian

Adiabaticity parameterAdiabaticity parameter

VtH

htJHk

jjjk

kz

jz

jk

))(1(

)(2

1

0

T/1

Adiabatic optimizationAdiabatic optimization

VHH )1()( 0 VHH )1()( 0

Approximate adiabatic Approximate adiabatic optimization vs. simulated optimization vs. simulated

annealingannealingVHH )1()( 0

Approximate adiabatic Approximate adiabatic optimization vs. simulated optimization vs. simulated

annealingannealing RMT theory near RMT theory near

centre of spectrum*centre of spectrum*Diffusive behaviourDiffusive behaviour

Residual energyResidual energy

ββ = 1 (GOE); 2 (GUE) = 1 (GOE); 2 (GUE)

Simulated Simulated annealing**annealing**

ζζ ≤≤ 6 6

*M. Wilkinson, PRA 41 (1990) 4645

**G.E. Santoro et al.,

Science 295 (2002) 2427

2/)2( TD

4/ TDT Tln

Running time vs. residual Running time vs. residual energyenergy

Classical/quantum simulated Classical/quantum simulated annealing (classical computer)annealing (classical computer)

Approximate adiabatic algorithm Approximate adiabatic algorithm (quantum computer)(quantum computer)

/1anneal exp T

/4adiab

T

Solution is encoded in the Solution is encoded in the final ground statefinal ground state

Error produces unusable Error produces unusable results (excited state does results (excited state does not, generally, encode an not, generally, encode an approximate solution)approximate solution)

Objective: minimize the Objective: minimize the probability of leavingprobability of leaving the the ground stateground state

Solution is a (smooth enough) Solution is a (smooth enough) function of the energy of the function of the energy of the final ground statefinal ground state

Error produces an approximate Error produces an approximate solution (energy of the excited solution (energy of the excited state is close to the ground state is close to the ground state energy)state energy)

Objective: minimize the Objective: minimize the average average driftdrift from the ground state from the ground state

Relevant problems: Relevant problems:

finding the ground state finding the ground state energy of a spin glassenergy of a spin glass

traveling salesman traveling salesman problemproblem

AQC vs. Approximate AQCAQC vs. Approximate AQC

Generic description of level Generic description of level evolution: Pechukas gas*evolution: Pechukas gas*

*P. Pechukas, PRL 51 (1983) 943

22,

3

2

0

11

2

)( ; ;

nkkmnmkknmkmn

nm nm

mnm

mm

mnnmmnmmmmm

mmm

xxxxlll

d

d

xx

lv

d

d

vxd

d

VEElVvEx

EVH

Pechukas gas kineticsPechukas gas kinetics

* * *

);,;,();,;,(

)()(

),(),(

* * *

)()()()()();,;,(

)()(

)()(),(

22

11

11

2

1

1

luyvxFluyvxf

lGlg

vxFvxf

llvuxyvvxxluyvxF

lllG

vvxxvxF

jkkkj

jj

jkjk

jjj

Pechukas gas kinetics: Pechukas gas kinetics: taking into account Landau-Zener taking into account Landau-Zener

transitionstransitions

1,

min2

4exp

mmLZ VP

Pechukas gas flow simulationPechukas gas flow simulation

Level collisions and LZ Level collisions and LZ transitionstransitions

““Diffusion” from the initial Diffusion” from the initial statestate

Analog vs. digitalAnalog vs. digital

4-flux qubit register4-flux qubit register

*M. Grajcar et al., PRL 96 (2006) 047006

ConclusionsConclusions

Eigenvalues behaviour is Eigenvalues behaviour is notnot described by simple described by simple diffusiondiffusion

Marginal states behaviour Marginal states behaviour qualitatively differentqualitatively different: : adiabatic evolution generally robustadiabatic evolution generally robust

Analog operation of quantum adiabatic computer Analog operation of quantum adiabatic computer provides exponential speedupprovides exponential speedup

Advantages of Pechukas mapping: exact, Advantages of Pechukas mapping: exact, provides intuitively clear description and provides intuitively clear description and controllable approximations (BBGKY chain)controllable approximations (BBGKY chain)

In future: external noise sources; mean-field In future: external noise sources; mean-field theory; quantitative theory of a specific algorithm theory; quantitative theory of a specific algorithm realization; investigation of the class of AA-realization; investigation of the class of AA-tractable problemstractable problems