Abstract
Density matrix quantum Monte Carlo (DMQMC) is a recently developed method for stochastically sampling the N-particle thermal density matrix to obtain exact-on-average energies for model and ab initio systems. We report a systematic numerical study of the sign problem in DMQMC based on simulations of atomic and molecular systems. In DMQMC, the density matrix is written in an outer product basis of Slater determinants. In principle, this means that DMQMC needs to sample a space that scales in the system size, N, as O[(exp(N))2]. In practice, removing the sign problem requires a total walker population that exceeds a system-dependent critical walker population (Nc), imposing limitations on both storage and compute time. We establish that Nc for DMQMC is the square of Nc for FCIQMC. By contrast, the minimum Nc in the interaction picture modification of DMQMC (IP-DMQMC) is only linearly related to the Nc for FCIQMC. We find that this difference originates from the difference in propagation of IP-DMQMC versus canonical DMQMC: the former is asymmetric, whereas the latter is symmetric. When an asymmetric mode of propagation is used in DMQMC, there is a much greater stochastic error and is thus prohibitively expensive for DMQMC without the interaction picture adaptation. Finally, we find that the equivalence between IP-DMQMC and FCIQMC seems to extend to the initiator approximation, which is often required to study larger systems with large basis sets. This suggests that IP-DMQMC offers a way to ameliorate the cost of moving between a Slater determinant space and an outer product basis.
Highlights
In a recent study, we showed that the density matrix quantum Monte Carlo (DMQMC) method could be applied to molecular systems, extending it beyond its original applications to model systems in condensed matter physics.[1]
DMQMC has been shown to be a promising method for finitetemperature applications, and in this work, we have confirmed that DMQMC shows the potential to be as effective for finite-temperature work as FCIQMC is for ground-state simulations
We found that the critical walker population in IP-DMQMC is the same as that in FCIQMC across all β values due to the asymmetric sampling present in IPDMQMC
Summary
We showed that the density matrix quantum Monte Carlo (DMQMC) method could be applied to molecular systems, extending it beyond its original applications to model systems in condensed matter physics.[1]. DMQMC joins a growing set of methods attempting to solve the finite-temperature problem that have attracted recent attention among quantum chemists, including other quantum Monte Carlo methods,[6−11] manybody theories,[12−14] and others.[15−19] Many of these methods, like DMQMC, continue to undergo development at the time of this publication.[20−28]. The original FCIQMC paper found that simulations exceeding a critical walker population were able to successfully resolve the sign of the wave function and generate an exact-on-average energy estimate; it was not possible to find accurate estimates of the energy from simulations containing populations lower than this plateau.[29] FCIQMC employs a discrete basis set, which means that walkers arriving at the same site can be exactly annihilated. When the population has grown above the plateau, the sign problem is resolved, and exact energies can be collected in a straightforward manner
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