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arXiv:2512.04056 · Physical review. B./Physical review. B

Sign-resolved statistics and the origin of bias in quantum Monte Carlo

Ryan Larson, Rubem Mondaini, Richard T. Scalettar

Published Apr 29, 2026• 0 citations
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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.

DOI
10.1103/x24t-bd2q

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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 S{\cal S}S of the weight leads to qualitatively wrong results -- most notably, an apparent suppression rather than enhancement of ddd-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 S=±1{\cal S}=\pm1S=±1, we derive an exact relation linking the bias from ignoring the sign to the difference between sign-resolved means, ΔμΔμΔμ, and the average sign, ⟨S⟩\langle {\cal S}\rangle⟨S⟩. Our framework provides a precise diagnostic of the origin of measurement bias in Quantum Monte Carlo and clarifies why observables such as the ddd-wave susceptibility are especially sensitive to the sign problem.
Auxiliary field Quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field xxx introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices Mσ(x)M_\sigma(x)Mσ​(x), where σ\sigmaσ labels the species, and hence is typically not positive definite. Indeed, the average sign ⟨S⟩\langle {\cal S} \rangle⟨S⟩ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of ⟨S⟩\langle {\cal S} \rangle⟨S⟩. Does ⟨S⟩→0\langle {\cal S} \rangle \rightarrow 0⟨S⟩→0 because {\it randomly} chosen field configurations have det(M(x))<0{\rm det}\big(M(x)\big) < 0det(M(x))<0, or does the `sign problem' arise as a result of {\it importance sampling} in the simulations? In the latter case, the process of weighting the configurations with ∣det(M(x))∣|{\rm det}\big(M(x)\big)|∣det(M(x))∣ might steer the simulation to a region of configuration space of xxx where positive and negative determinants are equally likely, even though randomly chosen xxx would preferentially have determinants with a single dominant sign. In this paper we address the relative importance of these two mechanisms for the vanishing of ⟨S⟩\langle {\cal S} \rangle⟨S⟩ in quantum simulations.
Auxiliary field quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field hhh introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices ${M}_{\ensuremath{\sigma}}(h)$, where $\ensuremath{\sigma}$ labels the species, and hence is typically not positive definite. Indeed, the average sign $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$. Does $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}\ensuremath{\rightarrow}0$ because randomly chosen field configurations have det[M(h)]<0\mathrm{det}[M(h)]<0det[M(h)]<0, or does the sign problem arise because the specific subset of configurations chosen by the weighting function have a greater preponderance of negative values? In the latter case, the process of weighting the configurations with ∣det[M(h)]∣|\mathrm{det}[M(h)]|∣det[M(h)]∣ might steer the simulation to a region of configuration space of hhh where positive and negative determinants are equally likely, even though randomly chosen hhh would preferentially have determinants with a single dominant sign. In this paper, we address the relative importance of these two mechanisms for the vanishing of $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ in quantum simulations.
The sign problem limits determinant quantum Monte Carlo studies of strongly correlated fermion systems. In the spin-channel Hubbard-Stratonovich decoupling, spin correlations are exactly related to auxiliary-field correlations. This relation implies that an antiferromagnetic transition reorganizes auxiliary-field configurations and thereby changes the statistical structure of the resulting fermion matrices. We use the adjacent gap ratio of low-lying singular values of the space-time fermion matrix to probe interaction-driven transitions in two half-filled honeycomb-lattice Hubbard models. In the sign-free honeycomb Hubbard model, the statistic tracks the established transition from a Dirac semimetal to an antiferromagnetic Mott insulator. In the complex-weight Haldane-Hubbard model, the transition-sensitive feature remains visible in the phase-quenched reference ensemble and occurs near previous estimates of the transition. Moreover, phase reweighting only weakly modifies the gap ratio over the regimes investigated, despite the rapid suppression of the average phase. These results establish singular-value statistics as a sign-problem-resilient probe of interaction-driven transitions in determinant quantum Monte Carlo.
The fermion sign problem, the biggest obstacle in quantum Monte Carlo calculations, is completely solved in this paper. Here, we find a strategy, in which the contribution from those negative-weighted paths is thoroughly cancelled or replaced by some positive-weighted paths. The crucial point lies on the Feynman path integral formula proposed in our group, which allows us to deeply analyze the Boltzmann weight of each path. Through mathematical proof, we demonstrate that physical quantities can be exactly calculated within a specific kind of paths, which have positive the Boltzmann weight. With this finding, a new Monte Carlo method is proposed, in which the fermion sign problem is absent. As an example, the current method is applied to the two-dimensional Hubbard model, and the results do manifest the correctness.
In this work, we study the correlation between entanglement and the negative sign problem in quantum Monte Carlo for the simulation of low-dimensional strongly correlated quantum many body systems. Entanglement entropy characterizes the difficulty of many-body simulation with tensor network state related methods, while the average sign measures the difficulty in many-body simulation for a variety of quantum Monte Carlo methods. Although there exist cases where one type of method works better than the other, it is desirable to find the possible correlation between entanglement and average sign for general hard strongly correlated systems regarding computational complexity. We take the doped two-dimensional Hubbard model as an example and numerically calculate the doping evolution of both the entanglement in the ground state with Density Matrix Renormalization Group and the average sign in the Auxiliary Field Quantum Monte Carlo simulation at low temperature. The results show that they are indeed correlated. The entanglement entropy (average sign) shows a peak (dip) around 20% doping, indicating that it is the difficult region for both methods. The vicinity of 20% doping is also the most intriguing region in both the Hubbard model and cuprate high-Tc superconductors where competing states with close energy intertwine with each other. Recognizing the correlation between entanglement and average sign provides new insight into our understanding of the difficulty in the simulation of strongly correlated quantum many-body systems.
Abstract The fermion sign problem, the biggest obstacle in quantum Monte Carlo calculations, is completely solved in this paper. Here, we find a strategy, in which the contribution from those negative-weighted paths is thoroughly cancelled or replaced by some positive-weighted paths. The crucial point lies on the Feynman path integral formula proposed in our group, which allows us to deeply analyze the Boltzmann weight of each path. Through mathematical proof, we demonstrate that physical quantities can be exactly calculated within a specific wavefunction space, in which all paths have positive Boltzmann weight. With this finding, a new Monte Carlo method is proposed, in which the fermion sign problem is absent. As an example, the current method is applied to the two-dimensional Hubbard model, and the results do manifest the correctness. *Corresponding author: dysun@phy.ecnu.edu.cn