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

TL;DR

The paper investigates the fermion sign problem in determinant quantum Monte Carlo by analyzing sign-resolved statistics of observables in the Hubbard model. By comparing sign-separated histograms P_+(O) and P_-(O), it derives an exact relation linking the measurement bias from ignoring the sign to the difference of sign-resolved means Δμ and the average sign ⟨S⟩ via ⟨O⟩_W − ⟨O⟩_{|W|} = Δμ(1−⟨S⟩^2)/(2⟨S⟩). The results show that while the sign-resolved histograms become similar at low temperature, the bias can still be substantial due to the amplification factor (1−⟨S⟩^2)/(2⟨S⟩) growing as ⟨S⟩→0, with the d-wave pairing susceptibility χ_d being particularly sensitive. The work provides a model-independent diagnostic framework, based on sign-resolved histograms and distributional distances (Wasserstein and Bhattacharyya), to quantify and understand the origin of bias in QMC measurements, and highlights the conditions under which sign-ignorant results may be qualitatively misleading. It also lays groundwork for evaluating sign problem severity across lattice geometries, models, and QMC schemes using observable histogram statistics rather than solely exponential signal decay.

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, $Δμ$, 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.

Figures (10)

• Figure 1: Probability density distributions of physical observables: (a) the kinetic energy $K$, (b) the antiferromagnetic structure factor $S_{\rm AF}$, and the $s^*$- and $d$-wave and pair susceptibilities, $\chi_{s^*}$ and $\chi_d$, in (c) and (d). These are resolved by the value of the sign of the corresponding weight, and vertical lines give the mean values, $\mu_\pm$. All data are computed for an $8\times 8$ spatial lattice with $U/t = 6$, $T/t=1/3$ and $\mu/t=-1.4$; this leads to a mean density $\langle \hat{n}\rangle \simeq 0.88$ with $\langle {\cal S} \rangle \simeq 0.83$. The imaginary-time discretization is set at $t\Delta\tau = 0.05$.
• Figure 2: Difference of the means $\Delta \mu_{\cal O} = \langle {\cal O}\rangle_+ - \langle {\cal O}\rangle_-$ in the $T$ vs. $\mu$ plane. The observables $\cal O$ are the same as in Fig. \ref{['fig:fig_1_mod']}: (a) the kinetic energy $K$, (b) the antiferromagnetic structure factor $S_{\rm AF}$, (c) the $s^*$-wave pair susceptibility $\chi_{s^*}$ and (d) the $d$-wave pair susceptibility $\chi_d$. The colored marker indicates the set of parameters chosen for Fig. \ref{['fig:fig_1_mod']}, and the colors are derived from a symmetric log scale centered at zero, for enhanced visualization, with a linear range extending to the red-colored tick label in the color bar. Where applicable, parameters are as in Fig. \ref{['fig:fig_1_mod']}.
• Figure 3: The Wasserstein distance $W_1$ between the positive and negative sign histograms at $U/t=6$, normalized by a measure of the standard deviation of the combined distribution of both signs $\sigma_{\rm tot}$; results are averaged over 24 independent Markov chains, and the error bars are the standard error of the means. The normalization leads to a dimensionless quantity that facilitates the comparison of different physical quantities, specifically the ones shown in Figs. \ref{['fig:fig_1_mod']} and \ref{['fig:fig_2_mod']}. The temperature selected in Fig. \ref{['fig:fig_1_mod']} is marked as a vertical dashed line. System sizes are $L=8$ (solid markers) and $L=16$ (empty markers) for $\mu = -1.4t$; the inset shows the same data on a linear vertical scale.
• Figure 4: (a) The difference between the means $\Delta \mu_{\chi_{\alpha}}$ of the $\alpha = d, s^*$-wave susceptibilities $\chi_d$ and $\chi_{s^*}$ for configurations that have an associated positive and negative weights as a function of temperature; the right axis shows the fraction that enters Eq. \ref{['eq:sign_no_sign_relation']}. (b) [(c)] The average $\chi_d$ [$\chi_{s^*}$] when computing via considering the sign of the weights (i.e., the reweighted average) and ignoring the sign; error bars stem from a jackknife analysis. The inset in (a) shows the right-hand side of Eq. \ref{['eq:sign_no_sign_relation']} for these two quantities, which is proportional to the difference of the curves in (b) and (c). Unlike previous cases, we tune the chemical potential $\mu$ here to achieve a total density $\rho \simeq 0.875$; other parameters are similar to previous figures.
• Figure S1: Histograms of (a) the double occupancy $\rho_{\uparrow\downarrow}=\langle \hat{n}_{\uparrow} \hat{n}_{\downarrow} \rangle$, and (b-d) the equal time pair structure factors $P_s, P_{s^*}, P_d$. Parameters are the same as in Fig. \ref{['fig:fig_1_mod']} of the main text: $8\times 8$ spatial lattice with $U/t = 6$, $T/t=1/3$ and $\mu/t=-1.4$; this leads to a mean density $\rho \simeq 0.88$ with $\langle {\cal S} \rangle \simeq 0.83$. The imaginary-time discretization is set at $t\Delta\tau = 0.05$.