Quantum Annealing and Thermalization: Insights from Integrability

Li, Fuxiang; Chernyak, Vladimir; Sinitsyn, Nikolai A.

doi:10.1103/physrevlett.121.190601

“…The same solution in Ref. [31] also shows that if we need only a partial search by allowing a fraction α ≪ 1 of mistakes, i.e., allowing αN spins pointing in a wrong direction, then it is sufficient to choose g ∼ 1/(N α), i.e., the computation time reduces by a factor ∼ 1/(N α), so in our notation τ a ∼ O(1/α).…”

Section: Tmentioning

confidence: 68%

“…The exact solution of this model was found in Ref. [31]. It says that the ground state is determined if g ≈ 1.…”

Section: Tmentioning

confidence: 96%

Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Yan,

Sinitsyn

2021

Preprint

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We point to the existence of an analytical solution to a general quantum annealing (QA) problem of finding low energy states of an arbitrary Ising spin Hamiltonian HI by implementing time evolution with a Hamiltonian H(t) = HI + g(t)Ht. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling g(t) and a specific mixing Hamiltonian Ht that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of HI, the solution shows the possibility of a considerable quantum speedup of finding the Ising ground state. We then compare predictions of our solution to results of numerical simulations, and argue that the solvable QA protocol produces the optimal performance in the limit of maximal complexity of the computational problem. Our solution demonstrates for the most complex spin glasses a power-law energy relaxation with the annealing time T and uncorrelated from HI annealing schedule. This proves the possibility for spin glasses of a faster than ∼ 1/ log β T energy relaxation.

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“…The same solution in Ref. [30] also shows that if we need only a partial search by allowing a fraction α ≪ 1 of mistakes, i.e., allowing αN spins pointing in a wrong direction, then it is sufficient to choose g ∼ 1/(N α), i.e., the computation time reduces by a factor ∼ 1/(N α), so in our notation…”

Section: Tmentioning

confidence: 72%

“…The exact solution of this model was found in Ref. [30]. It says that the ground state is determined if g ≈ 1.…”

Section: Tmentioning

confidence: 96%

Analytical solution for low energy state estimation by quantum annealing to arbitrary Ising spin Hamiltonian

Yan

¹

,

Sinitsyn

²

2021

Preprint

0

View full text Add to dashboard Cite

We point to the existence of an analytical solution to a general quantum annealing (QA) problem of finding low energy states of an arbitrary Ising spin Hamiltonian HI by implementing time evolution with a Hamiltonian H(t) = HI + g(t)Ht. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling g(t) and a specific mixing Hamiltonian Ht that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of HI, the solution shows the possibility of a considerable quantum speedup of finding the Ising ground state. We then compare predictions of our solution to results of numerical simulations, and argue that the solvable QA protocol produces the optimal performance in the limit of maximal complexity of the computational problem. Our solution demonstrates for the most complex spin glasses a power-law energy relaxation with the annealing time T and uncorrelated from HI annealing schedule. This proves the possibility for spin glasses of a faster than ∼ 1/logβT energy relaxation.

show abstract

“…The auxiliary Hamiltonian plays a role of the generator for the corresponding parameter. Several solvable models were treated to construct the auxiliary Hamiltonians and to demonstrate that the method is useful to describe the time evolution [22][23][24].…”

mentioning

confidence: 99%

Counterdiabatic Hamiltonians for multistate Landau-Zener problem

Nishimura

¹

,

Takahashi

²

2018

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We study the Landau-Zener transitions generalized to multistate systems. Based on the work by Sinitsyn et al. [Phys. Rev. Lett. 120, 190402 (2018)], we introduce the auxiliary Hamiltonians that are interpreted as the counterdiabatic terms. We find that the counterdiabatic Hamiltonians satisfy the zero curvature condition. The general structures of the auxiliary Hamiltonians are studied in detail and the time-evolution operator is evaluated by using a deformation of the integration contour and asymptotic forms of the auxiliary Hamiltonians. For several spin models with transverse field, we calculate the transition probability between the initial and final ground states and find that the method is useful to study nonadiabatic regime.

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Quantum Annealing and Thermalization: Insights from Integrability

Cited by 21 publications

References 30 publications

Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Analytical solution for low energy state estimation by quantum annealing to arbitrary Ising spin Hamiltonian

Counterdiabatic Hamiltonians for multistate Landau-Zener problem

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