Quantum annealing of the random-field Ising model by transverse ferromagnetic interactions

Suzuki, Sei; Nishimori, Hidetoshi; Suzuki, Masuo

doi:10.1103/physreve.75.051112

Cited by 41 publications

(43 citation statements)

References 17 publications

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“…These exponential computational complexities do not come as a surprise because Theorems 2.2, 2.4 and 2.5 all apply to any optimization problems written in the generic form (18), which includes the worst cases of most difficult problems. Similar arguments apply to SA [34].…”

Section: Computational Complexitymentioning

confidence: 99%

“…A recent numerical study shows the effectiveness of this type of quantum kinetic energy [18]. The additional transverse interaction widens the instantaneous energy gap between the ground state and the first excited state.…”

Section: Transverse Ferromagnetic Interactionsmentioning

confidence: 99%

“…The coefficient Γ(t) is then gradually and monotonically decreased toward 0, leaving eventually only the potential term H Ising . Accordingly the state vector |ψ(t) , which follows the real-time Schrödinger equation, is expected to evolve from the trivial initial ground state of the transverse-field term (19) to the non-trivial ground state of (18), which is the solution of the optimization problem. An important issue is how slowly we should decrease Γ(t) to keep the state vector arbitrarily close to the instantaneous ground state of the total Hamiltonian (20).…”

Section: Transverse Field Ising Modelmentioning

confidence: 99%

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Mathematical foundation of quantum annealing

Morita

Nishimori

2008

Journal of Mathematical Physics

Self Cite

316

282

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Quantum annealing is a generic name of quantum algorithms to use quantum-mechanical fluctuations to search for the solution of optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundation of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrödinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence both for the Schrödinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping.

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Section: Computational Complexitymentioning

confidence: 99%

Section: Transverse Ferromagnetic Interactionsmentioning

confidence: 99%

Section: Transverse Field Ising Modelmentioning

confidence: 99%

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Mathematical foundation of quantum annealing

Morita

Nishimori

2008

Journal of Mathematical Physics

Self Cite

316

282

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show abstract

“…It is a generic mechanism, and we expect it to manifest in more complex quantum models where quantum fluctuations are introduced by transverse spin-spin or multispin interaction terms [21][22][23] in addition to the transverse field. We emphasize that the speedup mechanism we find is not limited to the models that can be efficiently simulated with quantum Monte Carlo dynamics on classical computers [24,25] (such as the transverse-field Ising model).…”

Section: Introductionmentioning

confidence: 99%

Open-System Quantum Annealing in Mean-Field Models with Exponential Degeneracy

Kechedzhi

Smelyanskiy

2016

Phys. Rev. X

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Real-life quantum computers are inevitably affected by intrinsic noise resulting in dissipative nonunitary dynamics realized by these devices. We consider an open-system quantum annealing algorithm optimized for such a realistic analog quantum device which takes advantage of noise-induced thermalization and relies on incoherent quantum tunneling at finite temperature. We theoretically analyze the performance of this algorithm considering a p-spin model that allows for a mean-field quasiclassical solution and, at the same time, demonstrates the first-order phase transition and exponential degeneracy of states, typical characteristics of spin glasses. We demonstrate that finite-temperature effects introduced by the noise are particularly important for the dynamics in the presence of the exponential degeneracy of metastable states. We determine the optimal regime of the open-system quantum annealing algorithm for this model and find that it can outperform simulated annealing in a range of parameters. Large-scale multiqubit quantum tunneling is instrumental for the quantum speedup in this model, which is possible because of the unusual nonmonotonous temperature dependence of the quantum-tunneling action in this model, where the most efficient transition rate corresponds to zero temperature. This model calculation is the first analytically tractable example where open-system quantum annealing algorithm outperforms simulated annealing, which can, in principle, be realized using an analog quantum computer.

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“…Although many useful results have been obtained in this field, many problems still need to be resolved. The main challenge is to accelerate the speed of QA algorithms so that the annealing time grows not exponentially but polynomially with the size of the problem [1][2][3][4][5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Non-Hermitian quantum annealing in the ferromagnetic Ising model

Nesterov¹,

Zepeda²,

Berman³

2013

Phys. Rev. A

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We developed a non-Hermitian quantum optimization algorithm to find the ground state of the ferromagnetic Ising model with up to 1024 spins (qubits). Our approach leads to significant reduction of the annealing time. Analytical and numerical results demonstrate that the total annealing time is proportional to ln N , where N is the number of spins. This encouraging result is important in using classical computers in combination with quantum algorithms for the fast solutions of NPcomplete problems. Additional research is proposed for extending our dissipative algorithm to more complicated problems.

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Quantum annealing of the random-field Ising model by transverse ferromagnetic interactions

Cited by 41 publications

References 17 publications

Mathematical foundation of quantum annealing

Mathematical foundation of quantum annealing

Open-System Quantum Annealing in Mean-Field Models with Exponential Degeneracy

Non-Hermitian quantum annealing in the ferromagnetic Ising model

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