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arXiv:2509.13017 · Physical Review A

Mitigating the sign problem by quantum computing

Kwai-Kong Ng, Min-Fong Yang

Published Feb 25, 2026• 0 citations
arXivHTMLPDF

Abstract

The notorious sign problem severely limits the applicability of quantum Monte Carlo (QMC) simulations, as statistical errors grow exponentially with system size and inverse temperature. A recent proposal of a quantum-computing stochastic series expansion (qc-SSE) method suggested that the problem could be avoided by introducing constant energy shifts into the Hamiltonian. Here we critically examine this framework and show that it does not strictly resolve the sign problem for Hamiltonians with non-commuting terms. Instead, it provides a practical mitigation strategy that suppresses the occurrence of negative weights. Using the antiferromagnetic anisotropic XY chain as a test case, we analyze the dependence of the average sign on system size, temperature, anisotropy, and shift parameters. An operator contraction method is introduced to improve efficiency. Our results demonstrate that moderate shifts optimally balance sign mitigation and statistical accuracy, while large shifts amplify errors, leaving the sign problem unresolved but alleviated.

DOI
10.1103/s4nb-6fdq

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The notorious sign problem severely limits the applicability of quantum Monte Carlo (QMC) simulations, as statistical errors grow exponentially with system size and inverse temperature. A recent proposal of a quantum-computing stochastic series expansion (qc-SSE) method suggested that the problem could be avoided by introducing constant energy shifts into the Hamiltonian. Here we critically examine this framework and show that it does not strictly resolve the sign problem for Hamiltonians with non-commuting terms. Instead, it provides a practical mitigation strategy that suppresses the occurrence of negative weights. Using the antiferromagnetic anisotropic XY chain as a test case, we analyze the dependence of the average sign on system size, temperature, anisotropy, and shift parameters. An operator contraction method is introduced to improve efficiency. Our results demonstrate that moderate shifts optimally balance sign mitigation and statistical accuracy, while large shifts amplify errors, leaving the sign problem unresolved but alleviated.
Quantum Monte Carlo (QMC) methods are the gold standard for studying equilibrium properties of quantum many-body systems. However, in many interesting situations, QMC methods are faced with a sign problem, causing the severe limitation of an exponential increase in the runtime of the QMC algorithm. In this work, we develop a systematic, generally applicable, and practically feasible methodology for easing the sign problem by efficiently computable basis changes and use it to rigorously assess the sign problem. Our framework introduces measures of non-stoquasticity that-as we demonstrate analytically and numerically-at the same time provide a practically relevant and efficiently computable figure of merit for the severity of the sign problem. Complementing this pragmatic mindset, we prove that easing the sign problem in terms of those measures is generally an NP-complete task for nearest-neighbor Hamiltonians and simple basis choices by a reduction to the MAXCUT-problem.
Sign problem in quantum Monte Carlo (QMC) simulation appears to be an extremely hard yet interesting problem. In this article, we present a pedagogical overview on the origin of the sign problem in various quantum Monte Carlo simulation techniques, ranging from the world-line and stochastic series expansion Monte Carlo for boson and spin systems to the determinant and momentum-space quantum Monte Carlo for interacting fermions. We point out the basis dependency of the sign problem and summarize the progresses to cure, ease and even make use of the sign problem over the years, such as symmetry analysis of the underlying Hamiltonian, basis optimization in writting down the partition functions and many others. Moreover, we state that although traditional lore saying that in case of sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space, there are recent breakthroughs showing this is not always the case and based on the properties of the partition function for finite size systems, one could distinguish when the average sign has the usual exponential scaling and when it is bestowed with an algebraic scaling at the low temperature limit. Fermionic QMC simulations with such algebraic sign problems have been successfully carried out for extended Hubbard-type and quantum Moiré lattice models.
Quantum Monte Carlo (QMC) methods for the frustrated quantum spin systems occasionally suffer from the negative sign problem, which makes simulations exponentially harder for larger systems at lower temperatures and severely limits QMC's application across a wide range of spin systems. This problem is known to depend on the choice of representation basis. We propose a systematic approach for mitigating the sign problem independent of the given Hamiltonian or lattice structure. We first introduce the concept of negativity to characterize the severity of the negative sign problem. We then demonstrate the existence of a locally defined quantity, the L1 adaptive loss function, which effectively approximates negativity, especially in frustration-free systems. Using the proposed loss function, we demonstrate that optimizing the representation basis can mitigate the negative sign. This is evidenced by several frustration-free models and other important quantum spin systems. Furthermore, we compare the effectiveness of unitary transformations against the standard orthogonal transformation and reveal that unitary transformations can effectively mitigate the sign problem in certain cases.
Abstract A quantum implementation of the Stochastic Series Expansion (SSE) Monte Carlo method is proposed, and is shown to offer significant advantages over classical implementations of SSE. In particular, for problems where classical SSE encounters the sign problem, the cost of implementing a Monte Carlo iteration scales only linearly with system size in quantum SSE, while it may scale exponentially with system size in classical SSE. In cases where classical SSE can be efficiently implemented, quantum SSE still offers an advantage by allowing for more general observables to be measured.
Quantum computation offers a promising alternative to classical Monte Carlo methods by naturally avoiding sign problems through unitary evolution. A common misconception follows from this: because quantum simulation sidesteps the sign problem, it is often assumed to be an essentially perfect substitute for classical sampling. This is false---quantum measurements are intrinsically stochastic, and estimating observables often requires a number of samples that grows exponentially with system size, time separation, or inverse temperature, exactly as in the classical signal-to-noise problem. In this work, we analyze the origin and structure of signal-to-noise problems in quantum computation, drawing parallels with classical Monte Carlo, and demonstrate that such noise persists even in sign-problem-free regimes. We discuss strategies to mitigate it, in particular a quantum generalization of Stein control variates construction, in which unbiased estimators are built from operator commutators [X,A][X,A][X,A] and their discrete diagonal counterparts, and validate the approach with exactly solvable numerical examples that quantify the achievable variance reduction. Our results underscore that variance reduction, not merely sign-problem avoidance, is required to make quantum algorithms practically viable for many-body physics.
Quantum Monte Carlo simulations of quantum many-body systems are plagued by the Fermion sign problem. The computational complexity of simulating Fermions scales exponentially in the projection time β and system size. The sign problem is basis dependent and an improved basis, for fixed errors, leads to exponentially quicker simulations. We show how to use sign-free quantum Monte Carlo simulations to optimize over the choice of basis on large two-dimensional systems. We numerically illustrate these techniques decreasing the "badness" of the sign problem by optimizing over single-particle basis rotations on one- and two-dimensional Hubbard systems. We find a generic rotation which improves the average sign of the Hubbard model for a wide range of U and densities for L×4 systems. In one example improvement, the average sign (and hence simulation cost at fixed accuracy) for the 16×4 Hubbard model at U/t=4 and n=0.75 increases by exp[8.64(6)β]. For typical projection times of β⪆100, this accelerates such simulation by many orders of magnitude.