TY - JOUR
T1 - Convergence of quantum annealing with real-time Schrödinger dynamics
AU - Morita, Satoshi
AU - Nishimori, Hidetoshi
N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.
PY - 2007/6
Y1 - 2007/6
N2 - 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.
AB - 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.
KW - Adiabatic theorem
KW - Annealing schedule
KW - Optimization problem
KW - Quantum annealing
KW - Transverse-field ising model
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U2 - 10.1143/JPSJ.76.064002
DO - 10.1143/JPSJ.76.064002
M3 - Article
AN - SCOPUS:34547441517
VL - 76
JO - Journal of the Physical Society of Japan
JF - Journal of the Physical Society of Japan
SN - 0031-9015
IS - 6
M1 - 064002
ER -