  We present an extension of constrained-path auxiliary-field quantum Monte
Carlo (CP-AFQMC) for the treatment of correlated electronic systems coupled to
phonons. The algorithm follows the standard CP-AFQMC approach for description
of the electronic degrees of freedom while phonons are described in first
quantization and propagated via a diffusion Monte Carlo approach. Our method is
tested on the one- and two-dimensional Holstein and Hubbard-Holstein models.
With a simple semiclassical trial wavefunction, our approach is remarkably
accurate for $\omega/(2\text{d}t\lambda) < 1$ for all parameters in the
Holstein model considered in this study. In addition, we empirically show that
the autocorrelation time scales as $1/\omega$ for $\omega/t \lesssim 1$, which
is an improvement over the $1/\omega^2$ scaling of the conventional determinant
quantum Monte Carlo algorithm. In the Hubbard-Holstein model, the accuracy of
our algorithm is found to be consistent with that of standard CP-AFQMC for the
Hubbard model when the Hubbard $U$ term dominates the physics of the model, and
is nearly exact when the ground state is dominated by the electron-phonon
coupling scale $\lambda$. The approach developed in this work should be
valuable for understanding the complex physics arising from the interplay
between electrons and phonons in both model lattice problems and ab-initio
systems.
