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Sign problem in full configuration interaction quantum Monte Carlo: Linear and sublinear representation regimes for the exact wave function
Journal article   Open access   Peer reviewed

Sign problem in full configuration interaction quantum Monte Carlo: Linear and sublinear representation regimes for the exact wave function

James J Shepherd, Gustavo E Scuseria and James S Spencer
Physical review. B, Condensed matter and materials physics, Vol.90(15)
2014
DOI: 10.1103/PhysRevB.90.155130
url
https://doi.org/10.1103/PhysRevB.90.155130View
Published (Version of record) Open Access

Abstract

We investigate the sign problem for full configuration interaction quantum Monte Carlo (FCIQMC), a stochastic algorithm for finding the ground-state solution of the Schr\"odinger equation with substantially reduced computational cost compared with exact diagonalization. We find $k$-space Hubbard models for which the solution is yielded with storage that grows sublinearly in the size of the many-body Hilbert space, in spite of using a wave function that is simply a linear combination of states. The FCIQMC algorithm is able to find this sublinear scaling regime without bias and with only a choice of the Hamiltonian basis. By means of a demonstration we solve for the energy of a 70-site half-filled system (with a space of ${10}^{38}$ determinants) in 250 core hours, substantially quicker than the $\ensuremath{\sim}{10}^{36}$ core hours that would be required by exact diagonalization. This is the largest space that has been sampled in an unbiased fashion. The challenge for the recently developed FCIQMC method is made clear: Expand the sublinear scaling regime while retaining exact-on-average accuracy. We comment upon the relationship between this and the scaling law previously observed in the initiator adaptation (i-FCIQMC). We argue that our results change the landscape for the development of FCIQMC and related methods.

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