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Physical Review A

Observations on the statistical iteration of matrices

J. H. Hetherington

Published Nov 1, 1984• 118 citations
Publisher

Abstract

Monte Carlo calculations which rely on the statistical iteration of some operator can sometimes lead to results in which the variance grows as the iterations proceed. Alternatively the variance can be stable but the result incorrect. The effects are demonstrated by the Monte Carlo iteration of a 2\ifmmode\times\else\texttimes\fi{} 2 matrix and analyzed in detail for this simple case. In addition an algebraic formulation of a full Monte Carlo calculation with many simultaneous configurations including a method for keeping the number of configurations constant is given. With this formulation it is shown that the naive sampling of the wave function and naive estimate of the eigenvalue based on the growth in the number of configurations will be stable but biased. It is made plausible that for a sufficiently large number ($M$) of simultaneous configurations the naive method leads to a result which approaches the correct one as $M\ensuremath{\rightarrow}\ensuremath{\infty}$. It is shown that the correct average eigenvalue and eigenvector is a certain weighted average defined so that it avoids the problem of the growing variance and thus becomes more accurate as the chain is extended.

DOI
10.1103/physreva.30.2713

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Monte Carlo calculations which rely on the statistical iteration of some operator can sometimes lead to results in which the variance grows as the iterations proceed. Alternatively the variance can be stable but the result incorrect. The effects are demonstrated by the Monte Carlo iteration of a 2\ifmmode\times\else\texttimes\fi{} 2 matrix and analyzed in detail for this simple case. In addition an algebraic formulation of a full Monte Carlo calculation with many simultaneous configurations including a method for keeping the number of configurations constant is given. With this formulation it is shown that the naive sampling of the wave function and naive estimate of the eigenvalue based on the growth in the number of configurations will be stable but biased. It is made plausible that for a sufficiently large number (MMM) of simultaneous configurations the naive method leads to a result which approaches the correct one as M\ensuremath→\ensuremath∞M\ensuremath{\rightarrow}\ensuremath{\infty}M\ensuremath→\ensuremath∞. It is shown that the correct average eigenvalue and eigenvector is a certain weighted average defined so that it avoids the problem of the growing variance and thus becomes more accurate as the chain is extended.
Projection methods such as Green's function and diffusion Monte Carlo are commonly used to calculate the leading eigenvalue and eigenvector of operators or large matrices. They thereby give access to ground state properties of quantum systems, and finite temperature properties of classical statistical mechanical systems having a transfer matrix. The basis of these approaches is a stochastic application of the power method in which a "projection" operator is applied iteratively. For the systematic errors to be small, the number of iterations must be large; however, in that limit, the statistical errors grow tremendously. We present an analytical study of the main variance reduction methods used for dealing with this problem. In particular, we discuss the consequences of guiding, replication, and population control on statistical and systematic errors.
Despite the proven utility of quantum Monte Carlo methods in addressing the quantum many-body problem, many important observables are difficult to calculate due to the presence of large, and sometimes divergent, statistical errors. The present state of the art allows the construction of renormalized estimators which result in finite variances, but which invariably include some systematic bias. We present a simple method for calculating unbiased expectation values of local operators in the diffusion quantum Monte Carlo method which is applicable to both bare and renormalized estimators, allowing the accurate calculation of important properties such as forces.
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Variational Monte Carlo (VMC) is a powerful and fast-growing method for optimizing and evolving parameterized many-body wave functions, especially with modern neural-network quantum states. In practice, however, the stochastic estimators that form the backbone of the method can become unstable or biased due to the presence of nodes, a ubiquitous feature of quantum wave functions. In the continuum, this results in heavy-tailed estimators with potentially divergent variances, while in discrete Hilbert spaces the sampling distribution can miss parts of the support needed to form unbiased estimators. These statistical pathologies lead to unreliable optimization trajectories in stochastic reconfiguration or incorrect variational dynamics in time-dependent Variational Monte Carlo (t-VMC), and severely limit the power of the numerical simulations. We introduce blurred sampling to address these difficulties. The method has a number of rigorous properties that make it well-behaved, effective and computationally efficient. Additionally, it is a post-processing approach that can be used without modifying the underlying sampler and incurs only minimal overhead. We demonstrate its effectiveness in a broad range of problems -- including ground state optimization in both continuum and discrete settings, as well as in time-dependent variational Monte Carlo, where it resolves previously identified failure modes -- and further apply it to large-scale problems in spin dynamics. This work establishes a broadly applicable framework for robust VMC and t-VMC calculations.
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