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arXiv:2311.17333 · Journal of Computational Physics

Path integral molecular dynamics approximations of quantum canonical observables

Xin Huang, Petr Plecháč, Mattias Sandberg, Anders Szepessy

Published Nov 28, 2024• 1 citation
arXivHTMLPDF

Abstract

Mean-field molecular dynamics based on path integrals is used to approximate canonical quantum observables for particle systems consisting of nuclei and electrons. A computational bottleneck is the Monte Carlo sampling from the Gibbs density of the electron operator, which due to the fermion sign problem has a computational complexity that scales exponentially with the number of electrons. In this work, we construct an algorithm that approximates the mean-field Hamiltonian by path integrals for fermions. The algorithm is based on the determinant of a matrix with components built on Brownian bridges connecting permuted electron coordinates. The computational work for n electrons is O ( n 3 ) , which reduces the computational complexity associated with the fermion sign problem. We analyze a bias resulting from this approximation and provide a rough computational error indicator. It remains to rigorously explain the surprisingly high accuracy for high temperatures. The method becomes infeasible at low temperatures due to a large sample variance.

DOI
10.1016/j.jcp.2024.113625

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Mean-field molecular dynamics based on path integrals is used to approximate canonical quantum observables for particle systems consisting of nuclei and electrons. A computational bottleneck is the Monte Carlo sampling from the Gibbs density of the electron operator, which due to the fermion sign problem has a computational complexity that scales exponentially with the number of electrons. In this work, we construct an algorithm that approximates the mean-field Hamiltonian by path integrals for fermions. The algorithm is based on the determinant of a matrix with components built on Brownian bridges connecting permuted electron coordinates. The computational work for n electrons is O ( n 3 ) , which reduces the computational complexity associated with the fermion sign problem. We analyze a bias resulting from this approximation and provide a rough computational error indicator. It remains to rigorously explain the surprisingly high accuracy for high temperatures. The method becomes infeasible at low temperatures due to a large sample variance.
In general, Quantum Monte Carlo methods suffer from a sign problem when simulating fermionic systems. This causes the efficiency of a simulation to decrease exponentially with the number of particles and inverse temperature. To circumvent this issue, a nodal constraint is often implemented, restricting the Monte Carlo procedure from sampling paths that cause the many-body density matrix to change sign. Unfortunately, this high-dimensional nodal surface is not a priori known unless the system is exactly solvable, resulting in uncontrolled errors. We will discuss two possible routes to extend the applicability of finite-temperatue path integral Monte Carlo. First we extend the regime where signful simulations are possible through a novel permutation sampling scheme. Afterwards, we discuss a method to variationally improve the nodal surface by minimizing a free energy during simulation. Applications of these methods will include both free and interacting electron gases, concluding with discussion concerning extension to inhomogeneous systems.
We present a method for performing path integral molecular dynamics (PIMD) simulations for fermions and address its sign problem. PIMD simulations are widely used for studying many-body quantum systems at thermal equilibrium. However, they assume that the particles are distinguishable and neglect bosonic and fermionic exchange effects. Interacting fermions play a key role in many chemical and physical systems, such as electrons in quantum dots and ultracold trapped atoms. A direct sampling of the fermionic partition function is impossible using PIMD since its integrand is not positive definite. We show that PIMD simulations for fermions are feasible by employing our recently developed method for bosonic PIMD and reweighting the results to obtain fermionic expectation values. The approach is tested against path integral Monte Carlo (PIMC) simulations for up to seven electrons in a two-dimensional quantum dot for a range of interaction strengths. However, like PIMC, the method suffers from the sign problem at low temperatures. We propose a simple approach for alleviating it by simulating an auxiliary system with a larger average sign and obtaining an upper bound to the energy of the original system using the Bogoliubov inequality. This allows fermions to be studied at temperatures lower than would otherwise have been feasible using PIMD, as demonstrated in the case of a three-electron quantum dot. Our results extend the boundaries of PIMD simulations of fermions and will hopefully stimulate the development of new approaches for tackling the sign problem.
An approximate treatment of exchange in finite-temperature path integral Monte Carlo simulations for fermions has been proposed. In this method, some of the fine details of density matrix due to permutations have been smoothed over or averaged out by using the coarse-grained approximation. The practical usefulness of the method is tested for interacting fermions in a three dimensional harmonic well. The results show that, the present method not only reduces the sign fluctuation of the density matrix, but also avoid the fermion system collapsing into boson system at low temperatures. The method is substantiated to be exact when applied to free particles.
Correlated fermions are of high interest in condensed matter (Fermi liquids, Wigner molecules), cold atomic gases and dense plasmas. Here we propose a novel approach to path integral Monte Carlo (PIMC) simulations of strongly degenerate non-ideal fermions at finite temperature by combining a fourth-order factorization of the density matrix with antisymmetric propagators, i.e., determinants, between all imaginary time slices. To efficiently run through the modified configuration space, we introduce a modification of the widely used continuous space worm algorithm, which allows for an efficient sampling at arbitrary system parameters. We demonstrate how the application of determinants achieves an effective blocking of permutations with opposite signs, leading to a significant relieve of the fermion sign problem. To benchmark the capability of our method regarding the simulation of degenerate fermions, we consider multiple electrons in a quantum dot and compare our results with other ab initio techniques, where they are available. The present permutation blocking PIMC approach allows us to obtain accurate results even for N = 20 electrons at low temperature and arbitrary coupling, where no other ab initio results have been reported, so far.
Starting from a \emph{probabilistic numerics} approach to the Fermion sign problem in path integral Monte Carlo, we recast the arithmetic calculation of a Fermionic observable as a statistical inference problem. We develop approaches that learn the behaviour of Fermion exchange cycles binned by the conjugacy class of the permutation group (which we term `permutation family'). This extends the work of DuBois, Brown and Alder\cite{dubois2017overcoming} to inhomogeneous and more complex systems. Monte Carlo samples are used to train models for both the probability of a permutation family and the energy of this set of exchange permutations. The overall Fermionic energy is then directly inferred from these models, without using a direct ratio estimator on the Monte Carlo samples. By imposing physical understanding as inductive priors, we produce accurate and useful fits that remain robust even in regimes with severe sign problems. We generalise the linear (ideal-gas style) models of DuBois et al. with Bayesian priors that enforce the intuitive models of Feynman\cite{Feynman1953A} at their asymptotic limits. These linear models serve as the baseline for a Long Short-Term Memory (LSTM) neural network, which is tasked with learning only the residual many-body \emph{correlations} on top of the physical model. We develop active important sampling methods driven by these models, which direct the Monte Carlo chains toward undersampled permutation regions, to efficiently reduce the variance in the observable. We apply this framework to small experiments on benchmark systems: the spin-polarised uniform electron gas, and electrons in a 2D harmonic confining potential. In both cases we demonstrate that this inference-based framework can extract stable energies in regimes where direct Monte Carlo sampling fails due to the sign problem.
Abstract We provide a new paradigm for quantum simulation based on path integration that allows quantum speedups to be observed for problems that are more naturally expressed using the path integral formalism rather than the conventional Hamiltonian formalism. We provide two novel quantum algorithms based on Hamiltonian versions of the path integral formulation and another for Lagrangians of the form m 2 x ˙ 2 − V ( x ) , while avoiding the sign problems that preclude efficient Monte-Carlo algorithms for such problems. This Lagrangian path integral algorithm is based on a new rigorous derivation of a discrete version of the Lagrangian path integral. Our first Hamiltonian path integral method breaks up the paths into short timesteps. It is efficient under appropriate sparsity assumptions and requires a number of queries to oracles that give the eigenvalues and overlaps between the eigenvectors of the Hamiltonian terms that scales as t o ( 1 ) / ϵ o ( 1 ) for simulation time t and error ϵ . The second approach uses long-time path integrals for near-adiabatic systems and has query complexity that scales as O ( 1 / ϵ ) if the energy eigenvalue gaps and simulation time is sufficiently long. Finally, we show that our Lagrangian simulation algorithm requires a number of queries to an oracle that computes the discrete Lagrangian that scales for a system with η particles in D + 1 dimensions as O ~ ( η D t 2 / ϵ ) if V ( x ) is bounded and finite and the wavefunction obeys appropriate position and momentum cutoffs. This shows that Lagrangian dynamics can be efficiently simulated on quantum computers and opens up the possibility for quantum field theories for which the Hamiltonian is unknown to be efficiently simulated on quantum computers.