Convergence of the Cumulant Expansion and Polynomial-Time Algorithm for Weakly Interacting Fermions

TL;DR

This paper introduces a rigorous polynomial-time randomized algorithm for computing the log-partition function of weakly interacting fermions, extending cumulant expansion convergence proofs and utilizing a novel tree-structure analysis.

Contribution

It extends cumulant expansion convergence proofs to non-periodic systems and develops a new randomized importance sampling algorithm with provable polynomial runtime.

Findings

Algorithm achieves polynomial runtime in system size and precision.

Provides rigorous proof of convergence for cumulant expansion in non-periodic systems.

Introduces a tree-structure analysis for summation of Feynman diagrams.

Abstract

We propose a randomized algorithm to compute the log-partition function of weakly interacting fermions with polynomial runtime in both the system size and precision. Although weakly interacting fermionic systems are considered tractable for many computational methods such as the diagrammatic quantum Monte Carlo, a mathematically rigorous proof of polynomial runtime has been lacking. In this work we first extend the proof techniques developed in previous works for proving the convergence of the cumulant expansion in periodic systems to the non-periodic case. A key equation used to analyze the sum of connected Feynman diagrams, which we call the tree-determinant expansion, reveals an underlying tree structure in the summation. This enables us to design a new randomized algorithm to compute the log-partition function through importance sampling augmented by belief propagation. This approach differs from the traditional method based on Markov chain Monte Carlo, whose efficiency is hard to guarantee, and enables us to obtain a algorithm with provable polynomial runtime.