  We introduce methodologies for highly scalable quantum Monte Carlo
simulations of electron-phonon models, and report benchmark results for the
Holstein model on the square lattice. The determinant quantum Monte Carlo
(DQMC) method is a widely used tool for simulating simple electron-phonon
models at finite temperatures, but incurs a computational cost that scales
cubically with system size. Alternatively, near-linear scaling with system size
can be achieved with the hybrid Monte Carlo (HMC) method and an integral
representation of the Fermion determinant. Here, we introduce a collection of
methodologies that make such simulations even faster. To combat "stiffness"
arising from the bosonic action, we review how Fourier acceleration can be
combined with time-step splitting. To overcome phonon sampling barriers
associated with strongly-bound bipolaron formation, we design global Monte
Carlo updates that approximately respect particle-hole symmetry. To accelerate
the iterative linear solver, we introduce a preconditioner that becomes exact
in the adiabatic limit of infinite atomic mass. Finally, we demonstrate how
stochastic measurements can be accelerated using fast Fourier transforms. These
methods are all complementary and, combined, may produce multiple orders of
magnitude speedup, depending on model details.
