  We investigate a systematic statistical bias found in full configuration
quantum Monte Carlo (FCIQMC) that originates from controlling a walker
population with a fluctuating shift parameter and can become large in bosonic
systems. FCIQMC is a powerful statistical method for obtaining information
about the ground state of a sparse and abstract matrix. We show that the shift
estimator has the nice property of providing an upper bound for the exact
ground state energy and all projected energy estimators, while a variational
estimator is still an upper bound to the exact energy with substantially
reduced bias. A scalar model of the general FCIQMC population dynamics leads to
an exactly solvable It\^o stochastic differential equation. It provides further
insights into the nature of the bias and gives accurate analytical predictions
for delayed cross-covariance and auto-covariance functions of the shift energy
estimator and the walker number. The model provides a toe-hold on finding a
cure for the population control bias. We provide evidence for non-universal
power-law scaling of the population control bias with walker number in the
Bose-Hubbard model for various estimators of the ground state energy based on
the shift or on projected energies. For the specific case of the
non-interacting Bose-Hubbard Hamiltonian we obtain a full analytical prediction
for the bias of the shift energy estimator.
