Issues and Observations on Applications of the Constrained-Path Monte Carlo Method to Many-Fermion Systems

TL;DR

This paper examines the constrained-path Monte Carlo method for many-fermion systems, highlighting key differences from fixed-node methods, and discusses implications for energy estimation accuracy and methodological improvements.

Contribution

It clarifies the distinctions between constrained-path and fixed-node Monte Carlo methods and proposes ways to obtain reliable energy bounds.

Findings

The mixed estimator is not an upper bound in the constrained-path method.

Differences in state space and constraints affect the method's properties.

Proposed methods can produce energy upper bounds.

Abstract

We report several important observations that underscore the distinctions between the constrained-path Monte Carlo method and the continuum and lattice versions of the fixed-node method. The main distinctions stem from the differences in the state space in which the random walk occurs and in the manner in which the random walkers are constrained. One consequence is that in the constrained-path method the so-called mixed estimator for the energy is not an upper bound to the exact energy, as previously claimed. Several ways of producing an energy upper bound are given, and relevant methodological aspects are illustrated with simple examples.