  We construct numerical integrators for Hamiltonian problems that may
advantageously replace the standard Verlet time-stepper within Hybrid Monte
Carlo and related simulations. Past attempts have often aimed at boosting the
order of accuracy of the integrator and/or reducing the size of its error
constants; order and error constant are relevant concepts in the limit of
vanishing step-length. We propose an alternative methodology based on the
performance of the integrator when sampling from Gaussian distributions with
not necessarily small step-lengths. We construct new splitting formulae that
require two, three or four force evaluations per time-step. Limited,
proof-of-concept numerical experiments suggest that the new integrators may
provide an improvement on the efficiency of the standard Verlet method,
especially in problems with high dimensionality.
