Convergence theorems for quantum annealing

Morita, Satoshi; Nishimori, Hidetoshi

doi:10.1088/0305-4470/39/45/004

Cited by 55 publications

(57 citation statements)

References 29 publications

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“…In most numerical studies, stochastic methods are used. In this section, we investigate two types of quantum Monte Carlo methods and prove their convergence theorems, following [40].…”

Section: Convergence Condition Of Qa -Quantum Monte Carlo Evolutionmentioning

confidence: 99%

Mathematical foundation of quantum annealing

Morita

Nishimori

2008

Journal of Mathematical Physics

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302

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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“…In most numerical studies, stochastic methods are used. In this section, we investigate two types of quantum Monte Carlo methods and prove their convergence theorems, following [40].…”

Section: Convergence Condition Of Qa -Quantum Monte Carlo Evolutionmentioning

confidence: 99%

Mathematical foundation of quantum annealing

Morita

Nishimori

2008

Journal of Mathematical Physics

Self Cite

302

282

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“…Sufficient condition for the convergence of stochastic implementations of QA has been proved for the transverse-field Ising model [39]. A power-law decrease of the transverse field has been shown to be sufficient to guarantee the convergence to the optimal state for generic optimization problems.…”

Section: Theoretical Analysismentioning

confidence: 99%

Trust Evaluation via Large-Scale Complex Service-Oriented Online Social Networks

Liu

Jia

2015

IEEE Trans. Syst. Man Cybern, Syst.

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Service-oriented online social networks (SOSNs) are emerging ubiquitous platforms for numerous services where service consumers require the selection of trustworthy service providers who are unknown to them before invoking services with the aid of other intermediate participants. Under this circumstance, evaluation of the trust level of the service provider along the social trust paths from the service consumer to the service provider is required. To this end, selection of the optimal social trust path (OSTP) that can yield the most trustworthy evaluation result is a prerequisite. While existing single-trust-value methods can provide good but simple information to service consumers, more trust information, such as social intimacy degree between participants and role impact factor of intermediate participants, should be considered to represent the trust level of a service provider more comprehensively. When more trust information is considered, OSTP selection will become an NP-complete problem. In this paper, we propose path integral Monte Carlo quantum annealing (PIMCQA)-based OSTP (PIMCQA_OSTP) selection algorithm for complex SOSNs. PIMCQA_OSTP serves as the very first quantum inspired OSTP selection algorithm in complex SOSNs. Due to that quantum mechanics work with wave functions that can sample different regions of phase space equally well, and quantum systems can tunnel through classically impenetrable potential barriers between energy valleys, PIMCQA_OSTP shows its outstanding search ability and outperforms existing methods. Results of experiments on a real dataset of online social networks verify that PIMCQA_OSTP is a promising tool and is especially fit for complex SOSNs.Index Terms-Complex service-oriented online social networks, optimal social trust path selection (OSTP), path integral quantum annealing, quantum tunnel, trust level.

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“…Quantum mechanics was recently added to some optimization methods to overcome this difficulty. Instead of SA thermal fluctuations in a real space, quantum annealing (QA) uses quantum fluctuations and thus has a shorter convergence time [9], [10]. Quantum neural networks (QNNs) [11], [12] are variations of HNNs and were developed to effectively perform a full search on the basis of the superposition of quantum states.…”

Section: Introductionmentioning

confidence: 99%

Global Nonlinear Optimization Based on Wave Function and Wave Coefficient Equation

Satoh

2010

IEICE Trans. Fundamentals

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SUMMARY A method was developed for deriving the approximate global optimum of a nonlinear objective function with multiple local optimums. The objective function is expanded into a linear wave coefficient equation, so the problem of maximizing the objective function is reduced to that of maximizing a quadratic function with respect to the wave coefficients. Because a wave function expressed by the wave coefficients is used in the algorithm for maximizing the quadratic function, the algorithm is equivalent to a full search algorithm, i.e., one that searches in parallel for the global optimum in the whole domain of definition. Therefore, the global optimum is always derived. The method was evaluated for various objective functions, and computer simulation showed that a good approximation of the global optimum for each objective function can always be obtained.

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Convergence theorems for quantum annealing

Cited by 55 publications

References 29 publications

Mathematical foundation of quantum annealing

Mathematical foundation of quantum annealing

Trust Evaluation via Large-Scale Complex Service-Oriented Online Social Networks

Global Nonlinear Optimization Based on Wave Function and Wave Coefficient Equation

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