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Journal of the Physical Society of Japan

The Fermionic Minus-Sign Problem for the Ground State, Revisited with Higher Order Suzuki–Trotter Decompositions

Xiaoxing Wang, Hans‐Georg Matuttis

Published Sep 21, 2012• 2 citations

Abstract

A theorem by Yoshida which states that the discretization error for the Suzuki–Trotter (ST) decomposition is one order smaller than for the corresponding approximated Hamiltonian has long been overlooked with grave consequences for the accuracy of algorithms which use decomposition schemes. In our error analysis for the ground state energy of the Hubbard Hamiltonian by the projector quantum Monte Carlo method, we have used various orders of ST-decompositions, including for the first time in this field, pseudo-symplectic methods. We show that higher order does not necessarily imply better convergence, and identify the condition to obtain good convergence with higher order ST-decompositions. For the first order ST-decomposition, the ground state energy does not converge to the values for numerical diagonalization, in agreement with Yoshida's theorem, which may have caused confusion in connection with the “fermionic sign problem”. We show how the use of the sign of the fermion determinant is in fact a reweighing method which for the ground state energy violates some basic Monte Carlo properties (the Gaussian distribution of observables), in contrast to sampling without sign. Whether sampling with or without the sign deviates less from the exact ground state energy depends on the system parameters (interaction, filling) and on the test wave function, so sampling with sign gives not necessarily better or more physical than sampling without sign, a result which can be supported by arguments on the error compensation. We discuss the implications for related methods.

DOI: 10.1143/jpsj.81.104002

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A theorem by Yoshida which states that the discretization error for the Suzuki–Trotter (ST) decomposition is one order smaller than for the corresponding approximated Hamiltonian has long been overlooked with grave consequences for the accuracy of algorithms which use decomposition schemes. In our error analysis for the ground state energy of the Hubbard Hamiltonian by the projector quantum Monte Carlo method, we have used various orders of ST-decompositions, including for the first time in this field, pseudo-symplectic methods. We show that higher order does not necessarily imply better convergence, and identify the condition to obtain good convergence with higher order ST-decompositions. For the first order ST-decomposition, the ground state energy does not converge to the values for numerical diagonalization, in agreement with Yoshida's theorem, which may have caused confusion in connection with the “fermionic sign problem”. We show how the use of the sign of the fermion determinant is in fact a reweighing method which for the ground state energy violates some basic Monte Carlo properties (the Gaussian distribution of observables), in contrast to sampling without sign. Whether sampling with or without the sign deviates less from the exact ground state energy depends on the system parameters (interaction, filling) and on the test wave function, so sampling with sign gives not necessarily better or more physical than sampling without sign, a result which can be supported by arguments on the error compensation. We discuss the implications for related methods.

In this paper we compare numerical results for the ground state of the Hubbard model obtained by Quantum-Monte-Carlo simulations with results from exact and stochastic diagonalizations. We find good agreement for the ground state energy and superconducting correlations for both, the repulsive and attractive Hubbard model. Special emphasis lies on the superconducting correlations in the repulsive Hubbard model, where the small magnitude of the values obtained by Monte-Carlo simulations gives rise to the question, whether these results might be caused by fluctuations or systematic errors of the method. Although we notice that the Quantum-Monte-Carlo method has convergence problems for large interactions, coinciding with a minus sign problem, we confirm the results of the diagonalization techniques for small and moderate interaction strengths. Additionally we investigate the numerical stability and the convergence of the Quantum-Monte-Carlo method in the attractive case, to study the influence of the minus sign problem on convergence. Also here in the absence of a minus sign problem we encounter convergence problems for strong interactions.

We introduce a method that combines the power of both the lattice Green' function Monte Carlo (LGFMC) with the auxiliary field quantum Monte Carlo (AFQMC) techniques, and allows us to compute exact ground-state properties of the Hubbard model for $U\ensuremath{\lesssim}4t$ on finite clusters. Thanks to LGFMC, one obtains unbiased zero temperature results, not affected by the so-called Trotter approximation of the imaginary time propagator ${e}^{\ensuremath{-}H\ensuremath{\tau}}$. At the same time, the AFQMC formalism yields a remarkably fast convergence in $\ensuremath{\tau}$ before the fermion sign problem becomes prohibitive. As an application we report ground-state energies of the Hubbard model at

U/t=4

with up to 100 sites.

By precisely writing down the matrix element of the local Boltzmann operator (

e^(-{\tau}h

), where h is the paired operator formed by two interacting terms conjugate to each other), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. Our formula presents remarkable outcomes such that: 1) the Hubbard-Stratonovich transformation is not necessary, and the auxiliary field is no longer needed, which can improve the computational efficiency. 2) The new formula improves the accuracy of the Suzuki-Trotter decomposition. Our method also provides a potential theoretical framework for analyzing and solving the fermion sign problem. As an example, we studied the one-dimensional half-filled Hubbard model at a finite temperature. The obtained results were in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be used in various studies.

The quantum Monte Carlo method for the Hubbard model using the Hubbard-Stratonovich transformation is reexamined. It is observed that the value of the ground state energy for a 3×2 and 4×4 system is estimated correctly by ignoring the minus-sign of the negative sampling weight appearing in the simulation using the projector method. It is also observed that the average of sign is positive and the correlation between the energy and the sign is weak for these systems. The reasons why the ground state energy is estimated correctly is given on a phenomenological base.

Accurately simulating long-time dynamics of many-body systems is a challenge in both classical and quantum computing due to the accumulation of Trotter errors. While low-order Trotter-Suzuki decompositions are straightforward to implement, their rapidly growing error limits access to long-time observables. We present a framework for constructing efficient high-order Trotter-Suzuki schemes by identifying their structure and directly optimizing their parameters over a high-dimensional space. This method enables the discovery of new schemes with significantly improved efficiency compared to traditional constructions, such as those by Suzuki and Yoshida. Based on the theoretical efficiency and practical performance, we recommend two novel highly efficient schemes at

4^{\textrm{th}}

and

6^{\textrm{th}}

order. We also demonstrate the effectiveness of these decompositions on the Heisenberg model and the quantum harmonic oscillator, and find that for a fixed final time they perform better across the computational cost. Even when using large time steps, they surpass established low-order schemes like the Leapfrog. Finally, we investigate the in-practice performance of different Trotter schemes and find the decompositions with more uniform coefficients tend to feature improved error accumulation over long times. We have included this observation into our choice of recommended schemes.

The minus sign problem, as a problem of the non-positive-definite measure of the path integrals, is examined in the Monte Carlo simulation of the Hubbard model. A new correction method in terms of the ratio of two measured quantities is proposed to calculate physical properties. The momentum distribution and the spin correlation as well as the energy in the ground state are tested against the exactly diagonalized results. The results show that this ratio correction method is a promising prescription to reduce the minus sign problem.