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arXiv:1612.01871 · Physical review. E

Recommender engine for continuous-time quantum Monte Carlo methods

Li Huang, Yi‐feng Yang, Lei Wang

Published Mar 29, 2017• 30 citations
arXivHTMLPDF

Abstract

Recommender systems play an essential role in the modern business world. They recommend favorable items such as books, movies, and search queries to users based on their past preferences. Applying similar ideas and techniques to Monte Carlo simulations of physical systems boosts their efficiency without sacrificing accuracy. Exploiting the quantum to classical mapping inherent in the continuous-time quantum Monte Carlo methods, we construct a classical molecular gas model to reproduce the quantum distributions. We then utilize powerful molecular simulation techniques to propose efficient quantum Monte Carlo updates. The recommender engine approach provides a general way to speed up the quantum impurity solvers.

DOI
10.1103/physreve.95.031301

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Recommender systems play an essential role in the modern business world. They recommend favorable items such as books, movies, and search queries to users based on their past preferences. Applying similar ideas and techniques to Monte Carlo simulations of physical systems boosts their efficiency without sacrificing accuracy. Exploiting the quantum to classical mapping inherent in the continuous-time quantum Monte Carlo methods, we construct a classical molecular gas model to reproduce the quantum distributions. We then utilize powerful molecular simulation techniques to propose efficient quantum Monte Carlo updates. The recommender engine approach provides a general way to speed up the quantum impurity solvers.
Featuring detailed explanations of the major algorithms used in quantum Monte Carlo simulations, this is the first textbook of its kind to provide a pedagogical overview of the field and its applications. The book provides a comprehensive introduction to the Monte Carlo method, its use, and its foundations, and examines algorithms for the simulation of quantum many-body lattice problems at finite and zero temperature. These algorithms include continuous-time loop and cluster algorithms for quantum spins, determinant methods for simulating fermions, power methods for computing ground and excited states, and the variational Monte Carlo method. Also discussed are continuous-time algorithms for quantum impurity models and their use within dynamical mean-field theory, along with algorithms for analytically continuing imaginary-time quantum Monte Carlo data. The parallelization of Monte Carlo simulations is also addressed. This is an essential resource for graduate students, teachers, and researchers interested in quantum Monte Carlo techniques.
Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of ``correlated electron'' materials as auxiliary problems whose solution gives the ``dynamical mean-field'' approximation to the self-energy and local correlation functions. These applications require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms reviewed in this article meet this challenge. Derivations and descriptions of the algorithms are presented in enough detail to allow other workers to write their own implementations, discuss the strengths and weaknesses of the methods, summarize the problems to which the new methods have been successfully applied, and outline prospects for future applications.
Featuring detailed explanations of the major algorithms used in quantum Monte Carlo simulations, this is the first textbook of its kind to provide a pedagogical overview of the field and its applications. The book provides a comprehensive introduction to the Monte Carlo method, its use, and its foundations, and examines algorithms for the simulation of quantum many-body lattice problems at finite and zero temperature. These algorithms include continuous-time loop and cluster algorithms for quantum spins, determinant methods for simulating fermions, power methods for computing ground and excited states, and the variational Monte Carlo method. Also discussed are continuous-time algorithms for quantum impurity models and their use within dynamical mean-field theory, along with algorithms for analytically continuing imaginary-time quantum Monte Carlo data. The parallelization of Monte Carlo simulations is also addressed. This is an essential resource for graduate students, teachers, and researchers interested in quantum Monte Carlo techniques.
The self-learning Monte Carlo (SLMC) method speeds up the Monte Carlo simulation by designing and training an effective model to propose efficient global updates. We implement the continuous-time quantum Monte Carlo algorithm for quantum impurity models in the framework of SLMC. We introduce and train a diagram generating function (DGF) to model the probability distribution of field configurations in the continuous imaginary time at all orders of diagrammatic expansion. By using the DGF to propose global updates, we show that the self-learning continuous-time Monte Carlo method can significantly reduce the computational complexity of the simulation.
We present a new continuous-time solver for quantum impurity models such as those relevant to dynamical mean field theory. It is based on a stochastic sampling of a perturbation expansion in the impurity-bath hybridization parameter. Comparisons with Monte Carlo and exact diagonalization calculations confirm the accuracy of the new approach, which allows very efficient simulations even at low temperatures and for strong interactions. As examples of the power of the method we present results for the temperature dependence of the kinetic energy and the free energy, enabling an accurate location of the temperature-driven metal-insulator transition.
Quantum-enhanced probability distribution algorithms represent a significant advancement in computational modeling, particularly for complex probabilistic scenarios. This paper explores the potential of leveraging quantum mechanics to dramatically improve the accuracy and speed of Monte Carlo simulations, addressing the inherent limitations of classical approaches. The core claim is to develop a novel algorithm that utilizes quantum superposition and entanglement to efficiently explore the space of possible distributions, thereby reducing computational cost and enhancing performance. The proposed mechanism centers around exploiting quantum phenomena to overcome the computational bottlenecks encountered in traditional Monte Carlo methods, offering a paradigm shift in probabilistic modeling. This work will detail the theoretical foundations of the proposed algorithm, its implementation and testing with illustrative examples, and its potential impact on various applications.