  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 resolution of the identity and Cholesky
decomposition approaches, 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.
