Sampling strategies for expectation values within the Herman--Kluk approximation

TL;DR

This paper explores advanced sampling strategies for efficiently computing quantum expectation values using the Herman--Kluk semiclassical propagator, focusing on Monte Carlo quadrature techniques and their convergence properties.

Contribution

It introduces a novel observable-dependent sampling approach and provides theoretical convergence analysis for Monte Carlo estimators in high-dimensional quantum calculations.

Findings

Observable-dependent sampling reduces variance in Monte Carlo estimates.

Convergence conditions are established and validated numerically.

Methods are tested on harmonic and anharmonic systems with promising results.

Abstract

When computing quantum-mechanical observables, the ``curse of dimensionality'' limits the naive approach that uses the quantum-mechanical wavefunction. The semiclassical Herman--Kluk propagator mitigates this curse by employing a grid-free ansatz to evaluate the expectation values of these observables. Here, we investigate quadrature techniques for this high-dimensional and highly oscillatory propagator. In particular, we analyze Monte Carlo quadratures using three different initial sampling approaches. The first two, based either on the Husimi density or its square root, are independent of the observable whereas the third approach, which is new, incorporates the observable in the sampling to minimize the variance of the Monte Carlo integrand at the initial time. We prove sufficient conditions for the convergence of the Monte Carlo estimators and provide convergence error estimates. The analytical results are validated by numerical experiments in various dimensions on a harmonic oscillator and on a Henon-Heiles potential with an increasing degree of anharmonicity.