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arXiv:cond-mat/9904255 · Physical review. B, Condensed matter

Adaptive sampling approach to the negative-sign problem in the auxiliary-field quantum Monte Carlo method

Yoshihiro Asai

Published Oct 15, 2000• 4 citations
arXivHTMLPDF

Abstract

We propose a sampling method to calculate the ground state of interacting quantum systems. This approximate method, which we call the adaptive sampling quantum Monte Carlo (ASQMC) method, utilizes information from the short projection time density matrix derived from the Monte Carlo steps. The ASQMC method circumvents the negative-sign problem by introducing a conditional probability distribution function and guiding walkers to avoid a set of canceling fields. With the ASQMC method, the negative-sign ratio is greatly reduced. The ratio becomes zero in the limit $\ensuremath{\Delta}\ensuremath{\tau}$ goes to zero without prior knowledge of the exact ground state wave function. Comparisons with numerical results obtained by using other methods are made, and we observe that the ASQMC method gives accurate results with less than $0.5%$ error for the energy over wide regions of physical parameter values.

DOI
10.1103/physrevb.62.10674

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We propose a sampling method to calculate the ground state of interacting quantum systems. This approximate method, which we call the adaptive sampling quantum Monte Carlo (ASQMC) method, utilizes information from the short projection time density matrix derived from the Monte Carlo steps. The ASQMC method circumvents the negative-sign problem by introducing a conditional probability distribution function and guiding walkers to avoid a set of canceling fields. With the ASQMC method, the negative-sign ratio is greatly reduced. The ratio becomes zero in the limit \ensuremathΔ\ensuremathτ\ensuremath{\Delta}\ensuremath{\tau}\ensuremathΔ\ensuremathτ goes to zero without prior knowledge of the exact ground state wave function. Comparisons with numerical results obtained by using other methods are made, and we observe that the ASQMC method gives accurate results with less than 0.50.5%0.5 error for the energy over wide regions of physical parameter values.
The recently proposed full configuration interaction quantum Monte Carlo method allows access to essentially exact ground-state energies of systems of interacting fermions substantially larger than previously tractable without knowledge of the nodal structure of the ground-state wave function. We investigate the nature of the sign problem in this method and how its severity depends on the system studied. We explain how cancellation of the positive and negative particles sampling the wave function ensures convergence to a stochastic representation of the many-fermion ground state and accounts for the characteristic population dynamics observed in simulations.
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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=4U/t=4U/t=4 with up to 100 sites.
to develop quantum Monte Carlo (QMC) approaches for excited states and for more systematic and accurate treatment of ground states of strongly correlated sytems and heavier elements. We use the auxiliary-field quantum Monte Carlo (AFQMC) framework, which is a many-body total energy method that looks like many coupled mean-field calculations. I will discuss this method’s relations and dierences with traditional density-functional and quantum chemistry methods, and with other forms of QMC. The AFQMC method uses importance-sampled random walks in the space of nonorthogonal Slater determinants to project out the many-body ground state, or excited state. The random walks are constrained by a trial wave function (TWF) to remove the so-called sign problem, giving approximate (but non-perturbative) results. With singledeterminant TWFs from Hartree-Fock or density-functional theory (DFT), this approach can be thought of as a post-DFT method with no parameter. In this form, the accuracy is typically comparable to the quantum chemistry coupled-cluster CCSD(T) method near equilibrium geometries, and better than CCSD(T) when bonds are stretched. The computational cost of AFQMC scales as N 3 with characteristic size of the system. Its accuracy can be further improved at the cost of increased computing (releasing the constraint) or better TWF. With AFQMC, we can systematically target selected orbitals to correlate. This allows many-body calculations in solids with a frozen core, with down-folded Hamiltonians, or possibly embeded in a DFT calculations to extend scales. Results from a few on-going applications will be presented, including band structures in simple solids, magnetic phases in MnO, and Cobalt adsoption on graphene.
Systems of correlated quantum matter can be a steep challenge to any would-be method of solution. Matrix-product state (MPS)-based methods can describe 1D systems quasiexactly, but often struggle to retain sufficient bipartite entanglement to accurately approximate 2D systems already. Conversely, Quantum Monte Carlo (QMC) approaches, based on sampling a probability distribution, can generally approximate 2D and 3D systems with an error that decays systematically with growing sampling size. However, QMC can suffer from the so-called sign problem, that makes the approach prohibitively costly for many systems of interest, such as repulsively interacting fermions away from commensurate densities and frustrated systems. In this article, we introduce a new hybrid approach, that combines auxiliary-field QMC (AFQMC) with MPS-based algorithms. This hybrid technique removes or reduces the sign problem (depending on the specific model) while also needing to retain much lower bipartite entanglement than brute-force application of a MPS-algorithm, without the use of uncontrolled approximations. We present two use-cases of the algorithm that would be challenging or impossible to address with any other approach, and quantify the extent of any remaining sign problem.
We propose a novel algorithm for calculating the ground-state energy of quantum many-body systems by combining auxiliary-field quantum Monte Carlo (AFQMC) with tensor-train sketching. In AFQMC, a good trial wavefunction to guide the random walk is crucial for improving the sampling efficiency and controlling the sign problem. Our proposed method iterates between determining a new trial wavefunction in the form of a tensor train, derived from the current walkers, and using this updated trial wavefunction to anchor the next phase of AFQMC. Numerical results demonstrate that the algorithm is highly accurate for large spin systems. The overlap between the estimated trial wavefunction and the ground-state wavefunction also achieves high fidelity. We additionally provide a convergence analysis, highlighting how an effective trial wavefunction can reduce the variance in the AFQMC energy estimation. From a complementary perspective, our algorithm also extends the reach of tensor-train methods for studying quantum many-body systems.