  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. This bias can become the
dominant error when the sign problem is absent, e.g. 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, when the sign problem is
absent, 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.
