Hidetoshi Nishimori Tokyo Institute of Technology

In collaboration with Yuya Seki and Beatriz Seoane Y. Seki and H. Nishimori, Phys. Rev. E85, 051112 (2012)

B. Seoane and H. Nishimori, J. Phys. A45, 435301 (2012) Y. Seki and H. Nishimori, J. Phys. A48, 335301 (2015)

See also T. Kadowaki and H. Nishimori, Phys. Rev. E58, 5355 (1998)

Mean-field Analysis of Quantum Annealing with XX-type Terms

1

Problem

7,...) 5, ,3( 11

=

−= ∑=

pN

NHpN

i

ziσ ↑↑↑↑

Find the ground state of Ising model starting from paramagnet

Quantum annealing with transverse field

∑∑==

−−

−=N

i

xi

pN

i

zi s

NsNsH

11

)1(1)( σσ

↓↓↓+↓↑↑+↑↑↑=→→→→=−=== ∑ ......... )0( ,0:0 gHsti

xiσ

↑↑↑↑=

−=== ∑ gN

NHstp

i

zi 1)1( ,1: στ 1st order transition

at s=sc Jorg et al 2010

N2

2

Quantum para

Ordered state

1st order quantum transition ↑↑↑↑→→→→

s=0 s=1 sc

s=0 sc s=1

E

→→→→ ↑↑↑↑

aNeE −∝∆

aNeE

22)(

1∝

∆∝τ

1st order: exponentially long time; Hard to solve

aNb eNE

22)(

1<<∝

∆∝τ

2nd order: moderate time; Easy to solve

reduction

3

start goal

Phase transition

∑∑∑==

−−

−−

−=

N

i

xi

i

xi

pN

i

zi s

NN

NNssH

1

2

1)1(1)1(1 ),( σσλσλλ

Solution: antiferromagnetic XX interaction

Non-stoquastic

−−+−−−++−+

**00*

*

(start) ),0( :0 ∑−==i

xiHs σλ (goal) 1)1,1( :1

p

i

ziN

NHs

−=== ∑σλ

λ=1 λ=0

s=0

s=1

start

goal

1st order

4

∑∑∑==

−−

−−

−=

N

i

xi

i

xi

pN

i

zi s

NN

NNssH

1

2

1)1(1)1(1 ),( σσλσλλ

Solution: antiferromagnetic XX interaction

Non-stoquastic

−−+−−−++−+

**00*

*

(start) ),0( :0 ∑−==i

xiHs σλ (goal) 1)1,1( :1

p

i

ziN

NHs

−=== ∑σλ

p=3 p=5 p=11 5

Exponential vs polynomial rate of gap closing

λ=0.1 polynomial

p=3 p=5 p=11

λ=0.3, p=11 exponential aNeE

22)(

1∝

∆∝τ

bNE

∝∆

∝ 2)(1τ

6

Why is p=3 special?

++= 32)( bmammF

3=p

++= 42)( cmammF

5≥p

Landau free energy

Asymmetric (odd-term) contributions are weak.

7

pN

i

ziN

NH

−= ∑=1

1 σ

∑∑∑==

−−

−−

−=

N

i

xi

k

i

xi

pN

i

zi s

NN

NNssH

11)1(1)1(1 ),( σσλσλλ

Cheating?

)odd (

even) (2/1

∝−

−

k

kaNkXp e

Ngg

Overlap between ground states

4=k3=k

Bapst, Semerjian

aNXp eNgg −− >>∝ 2/1

2

pN

i

zip N

NH

−= ∑

=1

1 σ2

12

1

= ∑

=

N

i

xiX N

NH σ

Significant overlap, thus embedding the answer

k-body XX interactions

8

Accidental overlap

More complex problem: Random interactions

)1(

4,...) 3, ,2(

2121

2

21

121

1

1...

......Hopfield

±==

=−=

∑

∑

=

+−

<<<

ξξξξ

σσσ

µµ

µ

µkk

k

k

k

ii

p

ik

iii

zi

zi

iii

ziiii

NJ

kJH

Hopfield model: non-trivial ground state

Quantum annealing with XX interactions

∑∑=

−−

−+=

N

i

xi

i

xi s

NNHssH

1

2

Hopfield )1(1)1()( σσλλ

Finite p Same as the non-random case: can avoid 1st order.

9

Extensive p

Npk 04.0 ,2 ==zj

ji

ziijJ σσ∑

<

−

5 4, ,3 04.0 1 == − kNp k

zi

zi

iii

ziiii k

k

kJ σσσ

2

21

121...

...∑<<<

−

10

Why does the AF XX term work?

2

problem1)1(

+−− ∑∑

i

xi

i

xi N

cNsH σσ

Quantum effects: strong

11

−−−

−−−−−

*00*0

0**

• c=0: Stoquastic (fixed sign in off-diagonal) Can be mapped to classical Ising and simulated efficiently.

−−+−−−++−+

**00*

*• c>0: Non-stoquastic (both signs in off-diagonal) Difficult to efficiently simulate classically.

Conclusion

• 1st order transition 2nd order transition

by the antiferromagnetic XX interactions.

Exponential reduction in computation time.

• These are the first examples where an intrinsic quantum

speedup has been shown to exist in quantum annealing.

• “Intrinsic quantum speedup” means an exponential reduction

of computation time by non-stoquastic (classically unable to

simulate) terms.

12