Sign-Resolved Statistics and the Origin of Bias in Quantum Monte Carlo

TL;DR

This paper introduces a sign-resolved statistical analysis method in Quantum Monte Carlo to diagnose and understand the origin of bias caused by the fermion sign problem, especially affecting measurements like the $d$-wave susceptibility.

Contribution

It presents a novel framework analyzing sign-resolved observables to precisely diagnose measurement bias due to the fermion sign problem in Quantum Monte Carlo simulations.

Findings

Derived an exact relation linking bias to sign-resolved means and average sign.

Showed why certain observables are more sensitive to the sign problem.

Provided a diagnostic tool for understanding measurement bias in quantum simulations.

Abstract

Quantum simulations are a powerful tool for exploring strongly correlated many-body phenomena. Yet, their reach is limited by the fermion sign problem, which causes configuration weights to become negative, compromising statistical sampling. In auxiliary-field Quantum Monte Carlo calculations of the doped Hubbard model, neglecting the sign ${\cal S}$ of the weight leads to qualitatively wrong results -- most notably, an apparent suppression rather than enhancement of $d$-wave pairing at low temperature. Here we approach the problem from a different perspective: instead of identifying negative-weight paths, we examine the statistics of measured observables in a sign-resolved manner. By analyzing histograms of key quantities (kinetic energy, antiferromagnetic structure factor, and pair susceptibilities) for configurations with ${\cal S}=\pm1$, we derive an exact relation linking the bias from ignoring the sign to the difference between sign-resolved means, $\Delta\mu$, and the average sign, $\langle {\cal S}\rangle$. Our framework provides a precise diagnostic of the origin of measurement bias in Quantum Monte Carlo and clarifies why observables such as the $d$-wave susceptibility are especially sensitive to the sign problem.