Quantum annealing in a kinetically constrained system

Das, Arnab; Chakrabarti, Bikas K.; Stinchcombe, R B

doi:10.1103/physreve.72.026701

Cited by 36 publications

(37 citation statements)

References 14 publications

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“…(15). This result implies that the additional non-zero off-diagonal elements of HðtÞ would widen the energy gap and accelerate the convergence of QA.…”

mentioning

confidence: 95%

“…In this paper, we have derived conditions for convergence of QA under the real-time Schrödinger dynamics using the adiabatic theorem. The asymptotic power-law annealing schedule (15) guarantees the adiabatic evolution during the annealing process at its final stage t ) 1. This condition coincides with our previous results for stochastic implementations of QA.…”

mentioning

confidence: 99%

“…From this viewpoint, QA is also called quantum adiabatic evolution. 5) Most of the numerical studies [1][2][3][6][7][8][9][10][11][12][13][14][15] showed that QA is more efficient in solving optimization problems than the well-known classical algorithm, simulated annealing (SA). 16,17) Convergence theorems for stochastic implementations of QA have been proved for the transverse-field Ising model.…”

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confidence: 99%

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Convergence of Quantum Annealing with Real-Time Schrödinger Dynamics

Morita

Nishimori

2007

J. Phys. Soc. Jpn.

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Convergence conditions for quantum annealing are derived for optimization problems represented by the Ising model of a general form. Quantum fluctuations are introduced as a transverse field and/or transverse ferromagnetic interactions, and the time evolution follows the real-time Schrödinger equation. It is shown that the system stays arbitrarily close to the instantaneous ground state, finally reaching the target optimal state, if the strength of quantum fluctuations decreases sufficiently slowly, in particular inversely proportionally to the power of time in the asymptotic region. This is the same condition as the other implementations of quantum annealing, quantum Monte Carlo and Green's function Monte Carlo simulations, in spite of the essential difference in the type of dynamics. The method of analysis is an application of the adiabatic theorem in conjunction with an estimate of a lower bound of the energy gap based on the recently proposed idea of Somma et al. for the analysis of classical simulated annealing using a classical-quantum correspondence. Quantum annealing (QA) recently attracts much attention as a novel algorithm for optimization problems.1-4) A fictitious kinetic energy of quantum nature is introduced to the classical system which represents the cost function to be minimized. The resulting system searches the phase space by means of quantum transitions, which are gradually decreased as time proceeds. If the initial state is the ground state of the initial quantum Hamiltonian, the system is expected to keep track of the ground state of the instantaneous Hamiltonian under a slow decrease of quantum fluctuations. From this viewpoint, QA is also called quantum adiabatic evolution. 5) Most of the numerical studies [1][2][3][6][7][8][9][10][11][12][13][14][15] showed that QA is more efficient in solving optimization problems than the well-known classical algorithm, simulated annealing (SA). 16,17) Convergence theorems for stochastic implementations of QA have been proved for the transverse-field Ising model. 18)A power-law decrease of the transverse field has been shown to be sufficient to guarantee convergence to the optimal state for generic optimization problems. This power-law annealing schedule is faster than that of the inverse-log law for SA given in the theorem of Geman and Geman. 17,19) However, these theorems for QA were proved for stochastic processes to realize QA. It has been unknown so far what annealing schedule would guarantee the convergence of QA following the real-time Schrödinger equation. We have solved this problem on the basis of the idea of Somma et al. 20) These authors found that the inverse-log law condition for SA can be derived from the adiabatic theorem for a quantum system obtained from the original classical system through a classical-quantum mapping. Although they also discussed some aspects of QA, their interest was to use quantum mechanics to simulate finite-temperature classical statistical mechanics. We point out in the present article that the convergence co...

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“…(15). This result implies that the additional non-zero off-diagonal elements of HðtÞ would widen the energy gap and accelerate the convergence of QA.…”

mentioning

confidence: 95%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Convergence of Quantum Annealing with Real-Time Schrödinger Dynamics

Morita

Nishimori

2007

J. Phys. Soc. Jpn.

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“…In this Appendix, we prove the inequality (22). Although Hopf [32] originally proved this inequality for positive linear integral operators, we concentrate on a square matrix for simplicity.…”

Section: A Hopf's Inequalitymentioning

confidence: 99%

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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“…In such cases also one can employ quantum tunneling term to anneal the system. Das et al [11] have illustrated such a semiclassical treatment for kinetically constrained system (KCS) where they have shown QA is faster than SA. KCS is original East model with a kinetic constraint defined as the i-th spin can not flip if the (i − 1)-th spin is down.…”

Section: Introductionmentioning

confidence: 99%

Quantum annealing search of Ising spin glass ground state(s) with tunable transverse and longitudinal fields

Rajak

Chakrabarti

2014

Indian J Phys

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Here we first discuss briefly the quantum annealing technique. We then study the quantum annealing of Sherrington-Kirkpatrick spin glass model with the tuning of both transverse and longitudinal fields. Both the fields are time-dependent and vanish adiabatically at the same time, starting from high values. We solve, for rather small systems, the time-dependent Schrodinger equation of the total Hamiltonian by employing a numerical technique. At the end of annealing we obtain the final state having high overlap with the exact ground state(s) of classical spin glass system (obtained independently).

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Quantum annealing in a kinetically constrained system

Cited by 36 publications

References 14 publications

Convergence of Quantum Annealing with Real-Time Schrödinger Dynamics

Convergence of Quantum Annealing with Real-Time Schrödinger Dynamics

Mathematical foundation of quantum annealing

Quantum annealing search of Ising spin glass ground state(s) with tunable transverse and longitudinal fields

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