  The convergence of full configuration interaction quantum Monte Carlo
(FCIQMC) is accelerated using a quasi-Newton propagation (QN) which can also be
applied to coupled cluster Monte Carlo (CCMC). The computational scaling of
this optimised propagation is O(1), keeping the additional computational cost
to a bare minimum. Its effects are investigated deterministically and
stochastically on a model system, the uniform electron gas, with Hilbert space
size up to $10^{40}$ and shown to accelerate convergence of the instantaneous
projected energy by over an order of magnitude in the FCIQMC test case. Its
capabilities are then demonstrated with FCIQMC on an archetypical quantum
chemistry problem, the chromium dimer, in an all-electron basis set with
Hilbert space size of about $10^{22}$ yielding highly accurate FCI energies.
