nishimori

Quantum Annealing ― Basics and …

Hidetoshi Nishimori

Tokyo Institute of Technology

2

Motivation

Combinatorial optimization

∑−= jiijJH σσ

Ground state of Ising SG

Minimization of cost function : Multivariable (discrete) & Single-valued

3

Travelling salesman problem

Minimize the cost function (=tour length)

Configuration 1 Configuration 2 N N-1

N-2 N!

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Simulated Annealing (SA) • Generic, approximate algorithm

• Phase-space search by thermal fluctuations

tcNtTln

)( ≥t

cNtTln

)( ≥

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Quantum Annealing (QA) • Generic, approximate algorithm

• Phase-space search by quantum fluctuations

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Problems

¾Quantum vs thermal fluctuations:

Is QA useful for optimization purposes?

¾Related with quantum computation?

Yes, but be careful…

Yes, equivalent. Aharonov et al (2007)

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Implementation

∑∑ Γ−−=+= xi

zj

ziij tJHHtH σσσ )()( quantumclassical

t

Γ

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Experiments

∑ ∑Γ−−=i

xi

zj

ziijJH σσσ

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Magnetic material

Brooke, Bitko, Rosenbaum, Aeppli (1999)

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(Copyrighted material here)

Numerical evidence ∑ ∑Γ−−=

i

xi

zj

ziijJH σσσ

11

) (Finite ∑−= TJH jiij σσ

Amit, Gutfreunt, Sompolinsky (1985)

T vs Γ : Hopfield model

∑=

=p

jiijJ1µ

µµξξ

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(copyrighted material here)

)0( ∑ ∑ =Γ−−= TJHi

xi

zj

ziij σσσ

Nishimori & Nonomura (1996)

T vs Γ: Hopfield model

13


Frustrated system

Ferro interaction Antiferro interaction

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Master eqn vs. Schrödinger eqn

Kadowaki & Nishimori (1998)

Spin glass (SK model) with 8 spins

ttT 3)( =

tt 3)( =Γ

Schrödinger

Master /Equilibrium

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Monte Carlo for TSP

true)( :energy Residual EH −τ

Martonak, Santoro &Tosatti (2004)

quantumclassical 1)( HtHttH −+=ττ

cf: classical SA 0)( large)0( =⇒= τTT

classicalquantum )()0( HHHH =⇒= τ

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Monte Carlo for 3SAT

Battaglia, Santoro, Tosatti (2005)

QA

QA

SA

true?be Can

)falseor true(

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2142

3211

FCCF

xxxxCxxxC

i

∧==

∨∨=∨∨=

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Theoretical foundation ∑ ∑Γ−−=

i

xi

zj

ziijJH σσσ

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Convergence theorem

∑∑ Γ−−=+= xi

zj

ziij tJHHH σσσ )(quantumclassical

Nctt /)( −=Γ Morita & Nishimori

(Geman-Geman for SA)

Convergence condition

Control parameter

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)lnlnexp( ln

)(

)lnexp( )( /

δδ

δδ

bNtt

cNtT

aNttt Nc

=⇒==

=⇒==Γ −

Computational complexity

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Classical-quantum mapping

Classical equilibrium state Quantum ground state

ZeA

AH

T∑ −

=)()( σβσ )()( TATA ψψ=

0)( =THq ψ2/2/ )(1 HH

q eTMeH ββ −−=

Adiabatic condition (Quasi) equilibrium

tcNtTln

)( = Nctt /)( −=Γ

Somma, Batista & Ortiz (2007)

∑−

=σ

σβ

σψZ

eTzH 2/)(

)(

TFIM

22 22 22

Remark: Type of the dynamics

Nctt /)( −=Γ

Convergence condition (methods of proof)

9Real-time Schrödinger (adiabatic condition)

9Imaginary-time Schrödinger (adiabatic condition)

9Quantum Monte Carlo (reduction to SA by Suzuki-Trotter)

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Adiabatic evolution

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Quantum adiabatic evolution

t

E

zj

ziij

xi JtttH σσ

τσ

τ ∑∑ −−−= 1)(

Trivial initial state Non-trivial final state

∆ εψψ

=∆

∂∂

2

0

)(

)()(

t

tt

Htm

τ1

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Computational complexity

Adiabatic theorem 2−∆∝τ

∝∆−

−

b

aN

Ne

Gap scaling

Finite-size analysis

Complexity ∝(easy) (hard)

2

2

b

aN

Ne

τ

E

t

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View from quantum computation

Any problem hard classically but easy quantum mechanically?

Maybe (Fahri, Goldstone, Gutman, Lapan, Ludgrenm, Preda, 2001)

No (Young, Knysh, Smelyanskiy, 2010)

“Exact cover”

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“XORSAT”

No (Jörg, Krzakala, Semerjian, Zamponi, 2010)

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p-spin ferromagnet

( ) )/( 11

)(11

τσσ tssN

sNsHN

i

xi

pN

i

zi =−−−= ∑∑

==

Jörg, Krzakala, Kurchan, Maggs, Pujos (2010) “The problem that quantum annealing cannot solve”

• 1st order transition at finite s

• Exponentially small energy gap.

• Exponentially large time for adiabatic computation.

aNe−∝∆

bNe∝τ

0 1 s

QP F

Seki and Nishimori (2012) “The problem that quantum annealing CAN solve” 28

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An additional quantum term

( ) TFAFF0 )1()1(),( HsHHssH −+−+= λλλ2

1AFF

1= ∑

=

N

i

xiN

NH σ

Conventional case: λ=1

Start: s=0, λ=any Goal: s=１, λ=1

( ) TF0 )1( ),( HsHssH −+=λ

∑∑==

−=−=N

i

xi

pN

i

zi H

NNH

1TF

10 ,

1 σσ

s

λ 0

1

1

0 29

30 30 30 30 30 30 30 30 30 30

Result

1st 1st 2nd

p=3 p=5 p=11

( ) TFAFF0 )1()1(),( HsHHssH −+−+= λλλ30

Is essential?

kN

i

xi

k

NNH = ∑

=1

)(AFF

1 σ

( ) TF)(

AFF0 )1()1(),( HsHHssH k −+−+= λλλ

It works fine for k>2 as well.

Seoane and Nishimori (2012)

2

1AFF

1= ∑

=

N

i

xiN

NH σ

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Summary

32

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Summary

9QA works fine as a generic, approximate algorithm.

9“Better” than SA.

9Negative evidence for a few difficult problems.

9But there should be ways to avoid 1st order transitions.

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Collaborators

zTadashi Kadowaki

zHelmut G. Katzgraber

zYoshiki Matsuda

zSatoshi Morita

zYoshihiko Nonomura

zMasayuki Ohzeki

zMasuo Suzuki

zSei Suzuki

zYuya Seki

zBeatriz Seoane

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History

9Apolloni, Carvalho, de Falco (1989)

“QA”, algorithmic

9Finnila, Gomez, Sebenik, Stenson, Doll (1994)

Schrödinger, continuous (Lennard-Jones)

9Tanaka & Horiguchi (1997)

Image restoration, algorithmic

9Kadowaki & Nishimori (1998)

Transverse-field Ising, Schrödinger

9Fahri, Goldstone, Gutmann, Lapan, Lundgren, Preda (2001)

“Adiabatic computation”, complexity , not independent of KN

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