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arXiv:1004.2943 · Journal of Physics Conference Series

Deterministic flows of order-parameters in stochastic processes of quantum Monte Carlo method

Jun-ichi Inoue

Published Jun 1, 2010• 6 citations
arXivHTMLPDF

Abstract

In terms of the stochastic process of quantum-mechanical version of Markov chain Monte Carlo method (the MCMC), we analytically derive macroscopically deterministic flow equations of order parameters such as spontaneous magnetization in infinite-range ( d (= ∞ )-dimensional) quantum spin systems. By means of the Trotter decomposition, we consider the transition probability of Glauber-type dynamics of microscopic states for the corresponding ( d + 1)-dimensional classical system. Under the static approximation, differential equations with respect to macroscopic order parameters are explicitly obtained from the master equation that describes the microscopic-law. In the steady state, we show that the equations are identical to the saddle point equations for the equilibrium state of the same system. The equation for the dynamical Ising model is recovered in the classical limit. We also check the validity of the static approximation by making use of computer simulations for finite size systems and discuss several possible extensions of our approach to disordered spin systems for statistical-mechanical informatics. Especially, we shall use our procedure to evaluate the decoding process of Bayesian image restoration. With the assistance of the concept of dynamical replica theory (the DRT), we derive the zero-temperature flow equation of image restoration measure showing some 'non-monotonic' behaviour in its time evolution.

DOI
10.1088/1742-6596/233/1/012010

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In terms of the stochastic process of quantum-mechanical version of Markov chain Monte Carlo method (the MCMC), we analytically derive macroscopically deterministic flow equations of order parameters such as spontaneous magnetization in infinite-range ( d (= ∞ )-dimensional) quantum spin systems. By means of the Trotter decomposition, we consider the transition probability of Glauber-type dynamics of microscopic states for the corresponding ( d + 1)-dimensional classical system. Under the static approximation, differential equations with respect to macroscopic order parameters are explicitly obtained from the master equation that describes the microscopic-law. In the steady state, we show that the equations are identical to the saddle point equations for the equilibrium state of the same system. The equation for the dynamical Ising model is recovered in the classical limit. We also check the validity of the static approximation by making use of computer simulations for finite size systems and discuss several possible extensions of our approach to disordered spin systems for statistical-mechanical informatics. Especially, we shall use our procedure to evaluate the decoding process of Bayesian image restoration. With the assistance of the concept of dynamical replica theory (the DRT), we derive the zero-temperature flow equation of image restoration measure showing some 'non-monotonic' behaviour in its time evolution.
In terms of the stochastic process of a quantum-mechanical variant of Markov chain Monte Carlo method based on the Suzuki-Trotter decomposition, we an-alytically derive deterministic flows of order parameters such as magnetization in infinite-range (a mean-field like) quantum spin systems. Under the static approximation, differential equations with respect to order parameters are ex-plicitly obtained from the Master equation that describes the microscopic-law in the corresponding classical system. We discuss several possible applications of our approach to several research topics, say, image processing and neural networks. This paper is written as a self-review of two papers1,2 for Sympo-sium on Interface between Quantum Information and Statistical Physics at
We derive macroscopically deterministic flow equations with regard to the order parameters of the ferromagnetic p-spin model with infinite-range interactions. The p-spin model has a first-order phase transition for p>2. In the case of p≥5, the p-spin model with antiferromagnetic XX interaction has a second-order phase transition in a certain region. In this case, however, the model becomes a nonstoquastic Hamiltonian, resulting in a negative sign problem. To simulate the p-spin model with antiferromagnetic XX interaction, we utilize the adaptive quantum Monte Carlo method. By using this method, we can regard the effect of the antiferromagnetic XX interaction as fluctuations of the transverse magnetic field. A previous study [J. Inoue, J. Phys. Conf. Ser. 233, 012010 (2010)1742-659610.1088/1742-6596/233/1/012010] derived deterministic flow equations of the order parameters in the quantum Monte Carlo method. In this study, we derive macroscopically deterministic flow equations for the magnetization and transverse magnetization from the master equation in the adaptive quantum Monte Carlo method. Under the Suzuki-Trotter decomposition, we consider the Glauber-type stochastic process. We solve these differential equations by using the Runge-Kutta method, and we verify that these results are consistent with the saddle-point solution of mean-field theory. Finally, we analyze the stability of the equilibrium solutions obtained by the differential equations.
A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.
We propose a new efficient scheme for the quantum Monte Carlo study of quantum critical phenomena in quantum spin systems. Rieger and Young's Trotter-number-dependent finite-size scaling in quantum spin systems and Ito et al 's evaluation of the transition point with the non-equilibrium relaxation in classical spin systems are combined and generalized. That is, only one Trotter number and one inverse temperature proportional to system sizes are taken for each system size, and the transition point of the transformed classical spin model is estimated as the point at which the order parameter shows power-law decay. The present scheme is confirmed by the determination of the critical phenomenon of the one-dimensional asymmetric XY model in the ground state. The estimates of the transition point and the critical exponents , and are in good agreement with the exact solutions. The dynamical critical exponent is also estimated as , which is consistent with that of the two-dimensional Ising model.
We study the quantum version of the random K -satisfiability problem in the presence of an external magnetic field Gamma applied in the transverse direction. We derive the replica-symmetric free-energy functional within the static approximation and the saddle-point equation for the order parameter: the distribution P[h(m)] of functions of magnetizations. The order parameter is interpreted as the histogram of probability distributions of individual magnetizations. In the limit of zero temperature and small transverse fields, to leading order in Gamma magnetizations m approximately 0 become relevant in addition to purely classical values of m approximately +/-1 . Self-consistency equations for the order parameter are solved numerically using a quasi-Monte Carlo method for K=3 . It is shown that for an arbitrarily small Gamma quantum fluctuations destroy the phase transition present in the classical limit Gamma=0 , replacing it with a smooth crossover transition. The implications of this result with respect to the expected performance of quantum optimization algorithms via adiabatic evolution are discussed. The replica-symmetric solution of the classical random K -satisfiability problem is briefly reexamined. It is shown that the phase transition at T=0 predicted by the replica-symmetric theory is of continuous type with atypical critical exponents.
Abstract Quantum annealing aims to provide a faster method than classical computing for finding the minima of complicated functions, and it has created increasing interest in the relaxation dynamics of quantum spin systems. Moreover, problems in quantum annealing caused by first-order phase transitions can be reduced via appropriate temporal adjustment of control parameters, and in order to do this optimally, it is helpful to predict the evolution of the system at the level of macroscopic observables. Solving the dynamics of quantum ensembles is nontrivial, requiring modeling of both the quantum spin system and its interaction with the environment with which it exchanges energy. An alternative approach to the dynamics of quantum spin systems was proposed about a decade ago. It involves creating stochastic proxy dynamics via the Suzuki-Trotter mapping of the quantum ensemble to a classical one (the quantum Monte Carlo method), and deriving from this new dynamics closed macroscopic equations for macroscopic observables using the dynamical replica method. In this chapter, we give an introduction to this approach, focusing on the ideas and assumptions behind the derivations, and on its potential and limitations.