Quantum simulations are a powerful tool for exploring strongly correlated many-body phenomena. Yet, their reach is limited by the fermion sign problem, which renders configuration weights negative, thereby compromising statistical sampling. In auxiliary-field quantum Monte Carlo calculations of the doped Hubbard model, neglecting the sign $\mathcal{S}$ 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 $\mathcal{S}=\pm 1$, we derive an exact relation linking the bias from ignoring the sign to the difference between sign-resolved means, $\mathrm{\Delta }\mu$, and the average sign, $\langle \mathcal{S}\rangle$. Our framework provides a more precise diagnostic of why observables such as the $d$-wave susceptibility are especially sensitive to the sign problem.

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• Quantum many-body systems
• Hubbard model
• Many-body techniques
• Quantum Monte Carlo
• Statistical methods