Quantum Monte Carlo Sign-Resolved Statistics Reveal Bias Origin and Impact on Pairing Phenomena
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Quantum simulations represent a crucial method for investigating the complex behaviour of interacting quantum systems, but these calculations often encounter limitations due to the notorious fermion sign problem, which introduces inaccuracies in statistical sampling. Ryan Larson from the University of California, Davis, Rubem Mondaini from the University of Houston, and Richard T. Scalettar, also from the University of California, Davis, now present a new approach to understanding and diagnosing this issue. Rather than focusing on identifying problematic configurations, their work examines the statistics of measured properties, analysing data based on the sign of each configuration’s contribution.
The team derives a fundamental relationship connecting the resulting measurement bias to both the average sign and the difference between sign-resolved averages, offering a precise way to pinpoint the origin of errors in Monte Carlo calculations and explaining why certain properties, such as those related to electron pairing, are particularly vulnerable to the sign problem.
Results demonstrate that the sign problem arises from a complex interplay between local and global correlations, its severity depending strongly on the system’s filling and interaction strength. Specifically, competing interactions and frustration exacerbate the sign problem, leading to a rapid decay of the signal with increasing system size. Researchers examined the statistics of measured observables in a sign-resolved manner, establishing a precise link between the bias from ignoring the sign and the difference between sign-resolved means and the average sign. The core of their diagnostic lies in comparing the empirical distributions of observables measured on configurations with positive and negative signs. If the distributions for positive and negative signs coincide, ignoring the sign yields accurate results; conversely, differences indicate a biased estimate. Applying this approach to the Hubbard model with parameters relevant to cuprate materials, the study revealed noticeable differences in the histograms for key quantities, kinetic energy, antiferromagnetic structure factor, and pair susceptibilities. Specifically, the antiferromagnetic structure factor and d-wave pairing susceptibility exhibited distributions shifted to smaller values for negative signs, demonstrating that ignoring the sign introduces a significant bias in these observables. The research team examined the statistics of key observables, kinetic energy, antiferromagnetic structure factor, and pair susceptibilities, in a sign-resolved manner, analysing configurations with both positive and negative weights. This work reveals an exact relationship linking the bias from ignoring the sign of these configurations to the difference between sign-resolved means and the average sign. Experiments demonstrate that if histograms of measurements were identical for positive and negative sign sectors, ignoring the sign problem would yield accurate results for physical observables. However, the team found significant differences in these histograms, particularly for the d-wave pairing susceptibility, confirming that ignoring the sign introduces substantial errors. Measurements confirm that the bias incurred when the sign is ignored is directly connected to the difference of sign-resolved means and the average sign, as defined by a newly derived equation.
The team’s analysis of the Hubbard model, a widely studied model of metal-insulator transitions and superconductivity, shows that certain observables are particularly sensitive to the sign problem, with the d-wave pairing susceptibility especially affected, leading to qualitatively incorrect results when the sign is ignored. This work centres on analysing the statistics of measurable quantities in a sign-resolved manner, rather than attempting to identify problematic configurations with negative weights. By establishing a precise relationship linking measurement bias to the difference between sign-resolved means and the average sign, the team provides a diagnostic tool for understanding the origins of errors in Monte Carlo simulations. The findings clarify why certain observables, such as pairing susceptibilities, are particularly sensitive to the sign problem. The investigation demonstrates that the degree of dissimilarity between probability distributions obtained using positive weights and those incorporating both positive and negative weights can be quantified using various metrics, including the Wasserstein distance and the Bhattacharyya distance, yielding consistent results. Furthermore, the researchers derived a mathematical expression quantifying the bias incurred by ignoring the fermion sign, directly linking it to the difference between sign-resolved means and the average sign. Detailed analysis of histograms for key quantities, kinetic energy, antiferromagnetic structure factor, and pairing susceptibilities, resolved by both up and down spin signs, reveals the impact of the sign problem on different observables. This work provides a valuable theoretical foundation for improving the accuracy and reliability of quantum simulations, offering new insights into the behaviour of strongly correlated materials and other complex quantum systems. 👉 More information 🗞 Sign-Resolved Statistics and the Origin of Bias in Quantum Monte Carlo 🧠 ArXiv: https://arxiv.org/abs/2512.04056 Tags: