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ANALYSIS OF QUANTUM MONTE CARLO DYNAMICS IN INFINITE-RANGE ISING SPIN SYSTEMS: THEORY AND ITS POSSIBLE APPLICATIONS

Jun‐ichi Inoue

Published Sep 27, 2012• 0 citations
Publisher

Abstract

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

DOI
10.1142/9789814425285_0009

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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
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.
(Suzuki-)Trotter decompositions, splitting methods, (Lie) product formulae... The most common numerical methods for the time evolution of quantum systems come with many names. And they are used practically everywhere with applications ranging from the solution of classical equations of motion and various Monte Carlo simulations to the real and imaginary time evolution on classical as well as quantum computers. Here we review the state of the art of said methods, focussing especially on the progress made over the last few years. We highlight recently discovered efficient time evolution algorithms and explain how best to use them in practice. A central part of this work is the estimation of error bounds that has improved greatly within the past decade. The relevance of time evolution methods for quantum computing is discussed with a focus on noisy hardware. Finally, a comprehensive overview of generalisations, related methods and alternatives to Trotterization is provided. This includes time-dependent Hamiltonian dynamics, processed methods, multi-product formulae, symplectic integrators, TDVP for tensor networks, quantum signal processing, Crouch-Grossman methods and more. The overall perspective in this work is that of a theoretical physicist. All mathematical proofs as well as some technical details are omitted for easier readability. Instead, this review serves as a hands-on guide and, of course, as a starting point for references that provide further details.
Quantum annealing aims to provide a faster method for finding the minima of complicated functions, compared to classical computing, so there is an increasing interest in the relaxation dynamics of quantum spin systems. Moreover, it is known that problems in quantum annealing caused by first order phase transitions can be reduced via appropriate temporal adjustment of control parameters, aimed at steering the system away from local minima. To do this optimally, it would be helpful to predict the evolution of the system at the level of macroscopic observables. Solving the dynamics of a quantum ensemble is nontrivial, as it requires modelling not just the quantum spin system itself but also its interaction with the environment, with which it exchanges energy. An interesting alternative approach to the dynamics of quantum spin systems was proposed about a decade ago. It involves creating a 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.
Lately, there has been an extensive state-of-the-art research in Machine Learning methods, due to their important features such as universality in approximations and dimensional reduction. In this way, the present work aims at exploiting these properties of Machine Learning on physical many-body problems, in which a restricted Boltzmann Machine (RBM) is trained with quantum Monte Carlo data to best represent the ground state of an N spin system under the transverse field Ising model Hamiltonian in one dimension. During the path coursed along the development of this project, we first review classical well-implemented methods such as the single-layer perceptron, multi-layer perceptron for the XOR problem with back-propagation algorithm, discrete Hopfield network for pattern reconstruction, continuous Hopfield network with Simulated Annealing for solving the traveling salesman problem with 10 cities and the Boltzmann machine for feature extraction. Then, we review a few aspects of what is called a Quantum Boltzmann Machine and the basics of the transverse field Ising model. Finally, we train an RBM consisting of N visible neurons, hidden density neurons alpha=M/N=2 and 1-step contrastive divergence (CD_1), by minimizing the Kullbach-Liebler divergence of the RBM and the training data set coming from the algorithmic variant of quantum Monte Carlo, known as discrete Path-Integral Monte Carlo. For this last step, we use Trotter-Suzuki decomposition with m Trotter slices and the Wolff-s cluster algorithm. After training the RBM, we calculate magnetic observables based on the RBM-s wave-function such as longitudinal and transverse magnetization, for different values of transverse field and we do an estimation of the critical field at which the transverse Ising model system suffers a quantum phase transition from ferromagnet to paramagnet
Recent developments in computer simulations of phase transitions in Ising-like systems and thermodynamic behaviour of quantum spin chains are reviewed. A combination of stochastic Monte Carlo as well as deterministic transfer matrix and finite-size diagonalization methods is described in both fields as regards static properties. Asymptotic analysis and extrapolation techniques are presented in detail. Some effective-field methods, series expansions, spin dynamics simulations and experimental applications are also discussed.