Bilinear Quantum Monte Carlo: Expectations and Energy Differences

TL;DR

This paper introduces a bilinear sampling algorithm in Green's function Monte Carlo to accurately compute expectation values and energy differences for quantum systems, including non-commuting operators.

Contribution

It presents a novel bilinear Monte Carlo method that transforms Schrödinger equations into coupled equations, enabling precise estimations of quantum expectations and energy differences.

Findings

Successfully applied to model integral equations

Accurate energy difference calculations for hydrogen atom

Demonstrated effectiveness on test problems

Abstract

We propose a bilinear sampling algorithm in Green's function Monte Carlo for expectation values of operators that do not commute with the Hamiltonian and for differences between eigenvalues of different Hamiltonians. The integral representations of the Schroedinger equations are transformed into two equations whose solution has the form $\psi_a(x) t(x,y) \psi_b(y)$, where $\psi_a$ and $\psi_b$ are the wavefunctions for the two related systems and $t(x,y)$ is a kernel chosen to couple $x$ and $y$. The Monte Carlo process, with random walkers on the enlarged configuration space $x \otimes y$, solves these equations by generating densities whose asymptotic form is the above bilinear distribution. With such a distribution, exact Monte Carlo estimators can be obtained for the expectation values of quantum operators and for energy differences. We present results of these methods applied to several test problems, including a model integral equation, and the hydrogen atom.