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Quantum Monte Carlo (QMC) methods are often used to calculate properties of many body quantum systems. The main cost of many QMC methods, for example, the variational Monte Carlo (VMC) method, is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators so that the cost of constructing Slater matrices in these systems is now linear in the number of particles, whereas computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing the determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. The main contribution of this paper is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of incomplete LU decomposition with threshold and pivoting (ILUTP) preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of QMC algorithms from $O(n^3)$ per sweep to roughly $O(n^2)$, where $n$ is the number of particles, and a sweep is a sequence of $n$ steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors. Our results show that preconditioned iterative solvers can dramatically reduce the cost of VMC for large(r) systems.

Improved Scaling for Quantum Monte Carlo on Insulators