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arXiv:2112.05260 · Physical review. E

Mitigating spikes in fermion Monte Carlo methods by reshuffling measurements

Maksim Ulybyshev, Fakher F. Assaad

Published Aug 30, 2022• 6 citations
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Abstract

We propose a method to mitigate heavy-tailed distributions in fermion quantum Monte Carlo simulations originating from zeros of the fermion determinant. In this case, the second moment of the observables might not be well defined, and we show that by merely changing the synchronization between local updates and computation of observables, one can reduce the prefactor of the heavy-tailed distribution, thus substantially suppressing statistical fluctuations of observables. We also show that the average, or the first moment, is well defined and hence is independent on our measuring scheme. The method is especially suitable for local observables similar to, e.g., double occupancy, where the resulting speedup can reach two orders of magnitude. For observables containing spatial correlators, the speedup is more moderate but still ranges between five and ten. Our results are independent on the nature of the auxiliary field, discrete or continuous, and pave the way to improve measurement strategies for hybrid Monte Carlo simulations.

DOI
10.1103/physreve.106.025318

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We propose a method to mitigate heavy-tailed distributions in fermion quantum Monte Carlo simulations originating from zeros of the fermion determinant. In this case, the second moment of the observables might not be well defined, and we show that by merely changing the synchronization between local updates and computation of observables, one can reduce the prefactor of the heavy-tailed distribution, thus substantially suppressing statistical fluctuations of observables. We also show that the average, or the first moment, is well defined and hence is independent on our measuring scheme. The method is especially suitable for local observables similar to, e.g., double occupancy, where the resulting speedup can reach two orders of magnitude. For observables containing spatial correlators, the speedup is more moderate but still ranges between five and ten. Our results are independent on the nature of the auxiliary field, discrete or continuous, and pave the way to improve measurement strategies for hybrid Monte Carlo simulations.
We revisit the infinite variance problem in fermionic Monte Carlo simulations, which is widely encountered in areas ranging from condensed matter to nuclear and high-energy physics. The different algorithms, which we broadly refer to as determinantal quantum Monte Carlo (DQMC), are applied in many situations and differ in details, but they share a foundation in field theory, and often involve fermion determinants whose symmetry properties make the algorithm sign-problem-free. We show that the infinite variance problem arises as the observables computed in DQMC tend to form heavy-tailed distributions. To remedy this issue retrospectively, we introduce a tail-aware error estimation method to correct the otherwise unreliable estimates of confidence intervals. Furthermore, we demonstrate how to perform DQMC calculations that eliminate the infinite variance problem for a broad class of observables. Our approach is an exact bridge link method, which involves a simple and efficient modification to the standard DQMC algorithm. The method introduces no systematic bias and is straightforward to implement with minimal computational overhead. Our results establish a practical and robust solution to the infinite variance problem, with broad implications for improving the reliability of a variety of fundamental fermion simulations.
Fermionic Monte Carlo calculations with continuous auxiliary fields often encounter infinite variance problem from fermionic observables. This issue renders the estimation of observables unreliable, even with an infinite number of samples. In this work, we show that the infinite variance problem stems from the fermionic determinant. Also, we propose an approach to address this problem by employing a reweighting method that utilizes the distribution from an extra time-slice. Two strategies to compute the reweighting factor are explored: one involves truncating and analytically calculating the reweighting factor, while the other employs a secondary Monte Carlo estimation. With Hubbard model as a testbed, we demonstrate that utilizing the sub-Monte Carlo estimation, coupled with an unbiased estimator, offers a solution that effectively mitigates the infinite variance problem at a minimal additional cost.
Monte Carlo calculations of fermionic systems with continuous auxiliary fields frequently suffer from a diverging variance for fermionic observables. If the simulations have an infinite variance problem, one cannot estimate observables reliably even with an arbitrarily large number of samples. In this paper, we explore a method to solve this problem using sampling based on the distribution of a system with an extra time slice. The necessary reweighting factor is computed both perturbatively and through a secondary Monte Carlo. We show that the Monte Carlo reweighting coupled to the use of an unbiased estimator of the reweighting factor leads to a method that eliminates the infinite variance problem at a very small extra cost. We compute the double occupancy in the Hubbard model at half-filling to demonstrate the method and compare the results to well-established results obtained by other methods.
Pfaffian quantum Monte Carlo extends auxiliary field methods to anomalous fermionic Hamiltonians, opening access to pairing systems and Hubbard--Stratonovich decompositions that are unavailable in determinant formulations. We show that this flexibility can be combined with the global updates of Hybrid Monte Carlo (HMC), leading to an algorithm scaling as O(NτL3)\mathcal{O}(N_τL^3)O(Nτ​L3), similar to its determinant quantum Monte Carlo (DQMC) counterpart without pseudofermions. The central observation is that the molecular dynamics force depends only on the magnitude of the Pfaffian weight, which admits a determinant representation and leads to a simplified force expression. This construction also yields unequal-time Majorana correlators at essentially no additional cost. We demonstrate the method for the interacting Kitaev chain using the imaginary and symmetric Majorana hopping channel. The resulting noncommuting auxiliary-field bond operators are treated through an even-odd decomposition. Their block structure permits an analytic reduction to a sparse effective interaction matrix of bandwidth two and allows explicit force expressions. Results agree with exact diagonalization and resolve low-energy modes localized at opposite boundaries, together with their increasing overlap near the topological transition of the non-interacting system. These findings establish HMC as an efficient sampling framework for Pfaffian auxiliary-field simulations.
In the absence of a fermion sign problem, auxiliary-field (or determinantal) quantum Monte Carlo (DQMC) approaches have long been the numerical method of choice for unbiased, large-scale simulations of interacting many-fermion systems. More recently, the conceptual scope of this approach has been expanded by introducing ingenious schemes to compute entanglement entropies within its framework. On a practical level, these approaches, however, suffer from a variety of numerical instabilities that have largely impeded their applicability. Here we report on a number of algorithmic advances to overcome many of these numerical instabilities and significantly improve the calculation of entanglement measures in the zero-temperature projective DQMC approach, ultimately allowing us to reach similar system sizes as for the computation of conventional observables. We demonstrate the applicability of this improved DQMC approach by providing an entanglement perspective on the quantum phase transition from a magnetically ordered Mott insulator to a band insulator in the bilayer square lattice Hubbard model at half filling.
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.