quantum speedup by non-stoquastic operator
TRANSCRIPT
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 found out of the antiferromagnetic XX
interactions.
• “Intrinsic quantum speedup” means an exponential reduction
of computation time by non-stoquastic (classically unable to
simulate) terms. 12