Stochastic many-body perturbation theory for high-order calculations

TL;DR

This paper introduces PTQMC, a stochastic method for high-order many-body perturbation calculations that avoids exponential scaling and improves convergence assessment.

Contribution

The authors develop PTQMC, a novel stochastic approach for high-order perturbation theory that accurately computes coefficients and assesses wave-function complexity.

Findings

PTQMC reproduces MBPT coefficients up to 16th order.

Combining PTQMC with resummation techniques yields stable energy estimates.

The effective number of configurations $e^{S}$ indicates perturbation validity better than energy convergence.

Abstract

High-order perturbative $\textit{ab initio}$ calculations are challenging due to the rapidly growing configuration space and the difficulty of assessing convergence. In this letter, we introduce perturbation theory quantum Monte Carlo (PTQMC), a stochastic approach designed to compute high-order many-body perturbative corrections. By representing the perturbative wave function with random walkers in configuration space, PTQMC avoids the exponential scaling inherent to conventional constructions of high-rank excitation operators. Benchmark calculations for the Richardson pairing model demonstrate that PTQMC accurately reproduces exact many-body perturbation theory (MBPT) coefficients up to 16th order, even in strongly divergent regimes. We further show that combining PTQMC with series resummation techniques yields stable and precise energy estimates in cases where the straightforward perturbative series fails. Finally, we propose the effective number of configurations, $e^{S}$, as a global measure of perturbative wave-function complexity that can be directly extracted within PTQMC. We demonstrate that the saturation behavior of $e^{S}$ provides a more reliable indicator of the validity of perturbative expansions than energy convergence alone.