Novel quantum Monte Carlo technique with linear scaling.

Dario Alfè, Michael J. Gillan

We have reformulated the quantum Monte Carlo (QMC) technique so that a large part of the calculation scales linearly with the number of atoms. The reformulation is related to a recent alternative proposal for achieving linear-scaling QMC, based on maximally localized Wannier orbitals (MLWO), but has the advantage of greater simplicity. The technique we propose draws on methods recently developed for linear-scaling density functional theory. We report tests of the new technique on the insulator MgO, and show that its linear-scaling performance is somewhat better than that achieved by the MLWO approach. Implications for the application of QMC to large complex systems are pointed out.

Dario Alfè, Michael J. Gillan, ND Drummond

Michael Dzierzawa, X. Zotos

Andrew Williamson, Randolph Q. Hood, Jeffrey C. Grossman

A method is presented for using truncated, maximally localized Wannier functions to introduce sparsity into the Slater determinant part of the trial wave function in quantum Monte Carlo calculations. When combined with an efficient numerical evaluation of these localized orbitals, the dominant cost in the calculation, namely, the evaluation of the Slater determinant, scales linearly with system size. This technique is applied to accurate total energy calculation of hydrogenated silicon clusters and carbon fullerenes containing 20-1000 valence electrons.

Tommaso Parolini, Estelle M. Inack, Giuliano Giudici +1 more

Quantum tunneling is a valuable resource exploited by quantum annealers to solve complex optimization problems. Tunneling events also occur during projective quantum Monte Carlo (PQMC) simulations, and in a class of problems characterized by a double-well energy landscape their rate was found to scale linearly with the first energy gap, i.e., even more favorably than in physical quantum annealers, where the rate scales with the gap squared. Here we investigate how a guiding wave function---which is essential to make many-body PQMC simulations computationally feasible---affects the tunneling rate. The chosen test beds are a continuous-space double-well problem, the ferromagnetic quantum Ising chain, and the recently introduced shamrock model. As guiding wave function, we consider an approximate Boltzmann-type ansatz, the numerically exact ground state of the double-well model, and a neural-network wave function based on a Boltzmann machine. Remarkably, for each ansatz we find the same asymptotic linear scaling of the tunneling rate that was previously found in the PQMC simulations performed without a guiding wave function. We also provide a semiclassical theory for the double-well with exact guiding wave function that explains the observed linear scaling. These findings suggest that PQMC simulations guided by an accurate ansatz represent a valuable benchmark for physical quantum annealers and a potentially competitive quantum-inspired optimization technique.

Stephen Fahy