Schrödinger’s Elephants & Quantum Slide Rules A.M. Zagoskin (FRS RIKEN & UBC) S. Savel’ev (FRS...
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Transcript of Schrödinger’s Elephants & Quantum Slide Rules A.M. Zagoskin (FRS RIKEN & UBC) S. Savel’ev (FRS...
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