  We introduce a family of methods for the full configuration interaction
problem in quantum chemistry, based on the fast randomized iteration (FRI)
framework [L.-H. Lim and J. Weare, SIAM Rev. 59, 547 (2017)]. These methods,
which we term "FCI-FRI," stochastically impose sparsity during iterations of
the power method and can be viewed as a generalization of full configuration
interaction quantum Monte Carlo (FCIQMC) without walkers. In addition to the
multinomial scheme commonly used to sample excitations in FCIQMC, we present a
systematic scheme where excitations are not sampled independently. Performing
ground-state calculations on five small molecules at fixed cost, we find that
the systematic FCI-FRI scheme is 11 to 45 times more statistically efficient
than the multinomial FCI-FRI scheme, which is in turn 1.4 to 178 times more
statistically efficient than the original FCIQMC algorithm.
