Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2601.22559

Understanding the sign problem from an exact Path Integral Monte Carlo model of interacting harmonic fermions

Sa Chin

Published Jan 30, 2026• 0 citations
arXivHTMLPDF

Abstract

This work shows that the recently discovered operator contraction identity for solving the discreet Path Integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the Path Integral Monte Carlo energy at any time step for any number of fermions is known analytically, or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interaction can shift the sign problem to larger/smaller imaginary time but does not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in $D$ dimension, with $n=D+1$ fermion, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three-dimension. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and compared to results obtained by modern neural networks.

DOI
10.48550/arxiv.2601.22559

Related papers

Understanding the sign problem from an exact path integral Monte Carlo model of interacting harmonic fermions

Siu A. Chin

This work shows that the recently discovered operator contraction identity for solving the discrete path integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the path integral Monte Carlo energy at any time step for any number of fermions is known analytically or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interactions can shift the sign problem to larger/smaller imaginary time but do not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in D dimension, with n = D + 1 fermions, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three dimensions. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and are compared to results obtained by modern neural networks.

The Journal of Chemical Physics•Jun 11, 2026•0 citations

High-order path-integral Monte Carlo methods for solving quantum dot problems

Siu A. Chin

The conventional second-order path-integral Monte Carlo method is plagued with the sign problem in solving many-fermion systems. This is due to the large number of antisymmetric free-fermion propagators that are needed to extract the ground state wave function at large imaginary time. In this work we show that optimized fourth-order path-integral Monte Carlo methods, which use no more than five free-fermion propagators, can yield accurate quantum dot energies for up to 20 polarized electrons with the use of the Hamiltonian energy estimator.

arXiv:1411.6682•Physical Review E•Mar 11, 2015•42 citations

The Path Integral Monte Carlo Sign Problem Is Not Always NP-Hard: Harmonic Fermions Can Be Solved in Quadratic Time

Aarif Chaudhary, Jonas Valenzuela, Sa Chin

In the Path Integral Monte Carlo (PIMC) simulation of fermions in a harmonic trap, with and without pairwise harmonic interactions, the partition functions for any discrete number of imaginary time slices (or beads) and for any choice of the short-time propagator can be analytically obtained from the contracted determinant form of the propagator. This work shows that the resulting recursion relation can be reformulated in the $λ$-ring language, yielding a closed-form finite-bead partition function in two dimensions in terms of permutation statistics. This closed-form partition function can be evaluated by a special algorithm in $O(n^2)$ time, providing an exact and numerically stable scheme for reproducing the energies of the original (undoable) fermion PIMC simulation for $n=10^4$ or more fermions. This result provides a concrete framework in which the numerical instability of the sign problem is completely bypassed, and serves as a counterexample to the prevailing view that all truly fermionic PIMC sign problems are NP-hard.

arXiv:2609.09071•Sep 8, 2026•0 citations

Simple proof that there is no sign problem in path integral Monte Carlo simulations of fermions in one dimension

Siu A. Chin

It is widely known that there is no sign problem in path integral Monte Carlo (PIMC) simulations of fermions in one dimension. As far as the author is aware, there is no direct proof of this in the literature. This work shows that the sign of the N-fermion antisymmetric free propagator is given by the product of all possible pairs of particle separations, or relative displacements. For a nonvanishing closed-loop product of such propagators, as required by PIMC, all relative displacements from adjacent propagators are paired into perfect squares, and therefore the loop product must be positive, but only in one dimension. By comparison, permutation sampling, which does not evaluate the determinant of the antisymmetric propagator exactly, remains plagued by a low-level sign problem, even in one dimension.

arXiv:2407.01618•Physical review. E•Jun 25, 2024•3 citations

Fermionic sign problem minimization by constant path integral contour shifts

Christoph Gäntgen, Evan Berkowitz, Thomas Luu +2 more

The path integral formulation of quantum mechanical problems including fermions is often affected by a severe numerical sign problem. We show how such a sign problem can be alleviated by a judiciously chosen constant imaginary offset to the path integral. Such integration contour deformations introduce no additional computational cost to the Hamiltonian Monte Carlo algorithm, while its effective sample size is greatly increased. This makes otherwise unviable simulations efficient for a wide range of parameters. Applying our method to the Hubbard model, we find that the sign problem is significantly reduced. Furthermore, we prove that it vanishes completely for large chemical potentials, a regime where the sign problem is expected to be particularly severe without imaginary offsets. In addition to a numerical analysis of such optimized contour shifts, we analytically compute the shifts corresponding to the leading and next-to-leading order corrections to the action. We find that such simple approximations, free of significant computational cost, suffice in many cases. We present a simulation of ${\mathrm{C}}_{60}$ fullerenes (buckyballs) that are successful over a wide parameter range.

arXiv:2307.06785•Physical review. B./Physical review. B•May 20, 2024•8 citations

A pseudo-fermion propagator approach to the fermion sign problem

Yunuo Xiong, Hongwei Xiong

In this work, within the framework of path integral Monte Carlo, we construct a pseudo-fermion propagator by replacing the original fermionic determinant with its absolute value. This modified propagator defines an auxiliary system free from the fermion sign problem, enabling efficient simulations of fermionic systems. We found that by shifting the pseudo-fermion energy based on the energy of a non-interacting fermion system, we can efficiently and reliably infer the energy of fermionic systems in various situations, from strong quantum degeneracy to weak quantum degeneracy. We have performed numerical simulations of quantum dots confined in a two-dimensional harmonic potential and found excellent agreement with benchmark results provided by other established methods. We believe that this pseudo-fermion propagator framework opens up new possibilities for numerical simulations of fermionic systems.

arXiv:2508.09557•The Journal of Chemical Physics•Nov 3, 2025•3 citations
This work shows that the recently discovered operator contraction identity for solving the discreet Path Integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the Path Integral Monte Carlo energy at any time step for any number of fermions is known analytically, or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interaction can shift the sign problem to larger/smaller imaginary time but does not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in DDD dimension, with n=D+1n=D+1n=D+1 fermion, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three-dimension. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and compared to results obtained by modern neural networks.
This work shows that the recently discovered operator contraction identity for solving the discrete path integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the path integral Monte Carlo energy at any time step for any number of fermions is known analytically or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interactions can shift the sign problem to larger/smaller imaginary time but do not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in D dimension, with n = D + 1 fermions, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three dimensions. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and are compared to results obtained by modern neural networks.
The conventional second-order path-integral Monte Carlo method is plagued with the sign problem in solving many-fermion systems. This is due to the large number of antisymmetric free-fermion propagators that are needed to extract the ground state wave function at large imaginary time. In this work we show that optimized fourth-order path-integral Monte Carlo methods, which use no more than five free-fermion propagators, can yield accurate quantum dot energies for up to 20 polarized electrons with the use of the Hamiltonian energy estimator.
In the Path Integral Monte Carlo (PIMC) simulation of fermions in a harmonic trap, with and without pairwise harmonic interactions, the partition functions for any discrete number of imaginary time slices (or beads) and for any choice of the short-time propagator can be analytically obtained from the contracted determinant form of the propagator. This work shows that the resulting recursion relation can be reformulated in the λλλ-ring language, yielding a closed-form finite-bead partition function in two dimensions in terms of permutation statistics. This closed-form partition function can be evaluated by a special algorithm in O(n2)O(n^2)O(n2) time, providing an exact and numerically stable scheme for reproducing the energies of the original (undoable) fermion PIMC simulation for n=104n=10^4n=104 or more fermions. This result provides a concrete framework in which the numerical instability of the sign problem is completely bypassed, and serves as a counterexample to the prevailing view that all truly fermionic PIMC sign problems are NP-hard.
It is widely known that there is no sign problem in path integral Monte Carlo (PIMC) simulations of fermions in one dimension. As far as the author is aware, there is no direct proof of this in the literature. This work shows that the sign of the N-fermion antisymmetric free propagator is given by the product of all possible pairs of particle separations, or relative displacements. For a nonvanishing closed-loop product of such propagators, as required by PIMC, all relative displacements from adjacent propagators are paired into perfect squares, and therefore the loop product must be positive, but only in one dimension. By comparison, permutation sampling, which does not evaluate the determinant of the antisymmetric propagator exactly, remains plagued by a low-level sign problem, even in one dimension.
The path integral formulation of quantum mechanical problems including fermions is often affected by a severe numerical sign problem. We show how such a sign problem can be alleviated by a judiciously chosen constant imaginary offset to the path integral. Such integration contour deformations introduce no additional computational cost to the Hamiltonian Monte Carlo algorithm, while its effective sample size is greatly increased. This makes otherwise unviable simulations efficient for a wide range of parameters. Applying our method to the Hubbard model, we find that the sign problem is significantly reduced. Furthermore, we prove that it vanishes completely for large chemical potentials, a regime where the sign problem is expected to be particularly severe without imaginary offsets. In addition to a numerical analysis of such optimized contour shifts, we analytically compute the shifts corresponding to the leading and next-to-leading order corrections to the action. We find that such simple approximations, free of significant computational cost, suffice in many cases. We present a simulation of C60{\mathrm{C}}_{60}C60​ fullerenes (buckyballs) that are successful over a wide parameter range.
In this work, within the framework of path integral Monte Carlo, we construct a pseudo-fermion propagator by replacing the original fermionic determinant with its absolute value. This modified propagator defines an auxiliary system free from the fermion sign problem, enabling efficient simulations of fermionic systems. We found that by shifting the pseudo-fermion energy based on the energy of a non-interacting fermion system, we can efficiently and reliably infer the energy of fermionic systems in various situations, from strong quantum degeneracy to weak quantum degeneracy. We have performed numerical simulations of quantum dots confined in a two-dimensional harmonic potential and found excellent agreement with benchmark results provided by other established methods. We believe that this pseudo-fermion propagator framework opens up new possibilities for numerical simulations of fermionic systems.