The sign problem in full configuration interaction quantum Monte Carlo: Linear and sub-linear representation regimes for the exact wave function

TL;DR

This paper explores the sign problem in FCIQMC, demonstrating sub-linear scaling regimes for the wave function representation, enabling efficient solutions for large quantum systems with reduced computational resources.

Contribution

It identifies sub-linear scaling regimes in FCIQMC for certain models, providing insights to improve the method's efficiency and accuracy for large-scale quantum calculations.

Findings

Achieved sub-linear storage growth in specific Hubbard models.

Solved for a 70-site system in 250 core hours, vastly faster than exact diagonalization.

Demonstrated the largest unbiased sampling of such a large quantum space.

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 diagonalisation. We find $k$-space Hubbard models for which the solution is yielded with storage that grows sub-linearly in the size of the many-body Hilbert space, in spite of using a wave function that is simply linear combination of states. The FCIQMC algorithm is able to find this sub-linear scaling regime without bias and with only a choice of 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 $\sim$10$^{36}$ core hours that would be required by exact diagonalisation. 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 sub-linear scaling regime whilst retaining exact on average accuracy. This result rationalizes the success of the initiator adaptation (i-FCIQMC) and offers clues to improve it. We argue that our results changes the landscape for development of FCIQMC and related methods.