Optimization in random field Ising models by quantum annealing

Sarjala, Matti; Petäjä, Viljo; Alava, Mikko J.

doi:10.1088/1742-5468/2006/01/p01008

Cited by 17 publications

(21 citation statements)

References 24 publications

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“…We show that the power decay (21) satisfies the adiabaticity condition (17) which guarantees convergence to the ground state of H Ising as t → ∞. .…”

Section: Transverse Field Ising Modelmentioning

confidence: 88%

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Mathematical foundation of quantum annealing

Morita

Nishimori

2008

Journal of Mathematical Physics

313

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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“…We show that the power decay (21) satisfies the adiabaticity condition (17) which guarantees convergence to the ground state of H Ising as t → ∞. .…”

Section: Transverse Field Ising Modelmentioning

confidence: 88%

“…Here a and c are constants of O(N 0 ) and δ is a small parameter to control adiabaticity appearing in (17).…”

Section: Transverse Field Ising Modelmentioning

confidence: 99%

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Mathematical foundation of quantum annealing

Morita

Nishimori

2008

Journal of Mathematical Physics

313

282

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“…Introduction of transverse ferromagnetic interactions does not necessarily improve the result for the cases of disordered ground states. This observation is to be compared with the finding of Sarjala et al [13] who showed by quantum Monte Carlo simulations that quantum annealing by the conventional method is less efficient than simulated annealing when the ground state is strongly ferromagnetic. We may conclude that their consequence does not necessarily reflect intrinsic features of quantum annealing.…”

Section: B Resultsmentioning

confidence: 85%

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

2007

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We introduce transverse ferromagnetic interactions, in addition to a simple transverse field, to quantum annealing of the random-field Ising model to accelerate convergence toward the target ground state. The conventional approach using only the transverse-field term is known to be plagued by slow convergence when the true ground state has strong ferromagnetic characteristics for the random-field Ising model. The transverse ferromagnetic interactions are shown to improve the performance significantly in such cases. This conclusion is drawn from the analyses of the energy eigenvalues of instantaneous stationary states as well as by the very fast algorithm of Bethe-type mean-field annealing adopted to quantum systems. The present study highlights the importance of a flexible choice of the type of quantum fluctuations to achieve the best possible performance in quantum annealing. The existence of such flexibility is an outstanding advantage of quantum annealing over simulated annealing.

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“…The logarithmic scaling was recently supported by the quantum Monte-Carlo simulation [7], though the evolution of state in quantum Monte-Carlo is different from the one ruled by the Schrödinger equation. In contrast, the authors of this paper have shown another scaling law of residual energy given by…”

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confidence: 95%

Study on Quantum Annealing Using the Density Matrix Renormalization Group

Suzuki

Okada

2007

IIS

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Quantum annealing is a quantum algorithm proposed recently for combinatorial optimization problems. It manipulates time evolution of a quantum mechanical state and obtain an approximate solution. In order to implement the algorithm in classical computers, we propose to apply the density matrix renormalization group method. Simulation of the time evolution of a quantum mechanical state becomes possible by the density matrix renormalization group method for problems of large size. We explain quantum annealing and the density matrix renormalization group method, and present results of numerical simulation using them.

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Optimization in random field Ising models by quantum annealing

Cited by 17 publications

References 24 publications

Mathematical foundation of quantum annealing

Mathematical foundation of quantum annealing

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

Study on Quantum Annealing Using the Density Matrix Renormalization Group

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