Reduced scaling Hilbert space variational Monte Carlo

We show that for both single-Slater-Jastrow and Jastrow geminal power wave functions the formal cost scaling of Hilbert space variational Monte Carlo can be reduced from fifth to fourth order in the system size, thus bringing it in line with the long-standing scaling of its real space counterpart. While traditional quantum chemistry methods can reduce costs related to the two-electron integral tensor through various tensor decomposition methods, we show that such approaches are ineffective in the presence of Hilbert space Jastrow factors. Instead, we develop a simple semi-stochastic approach that can take similar advantage of the near-sparsity of this four-index tensor. Through demonstrations on alkanes of increasing length, we show that accuracy and overall statistical uncertainty are not meaningfully affected and that a total cost crossover is reached as early as 50 electrons when using a minimal basis. Further study will be needed to assess where the crossover occurs in more compact molecular geometries and larger basis sets and to explore how in that context the crossover can be accelerated.