  In this work, we report, for the first time, an implementation of fermionic
auxiliary-field quantum Monte Carlo (AFQMC) using matrix product state (MPS)
trial wavefunctions, dubbed MPS-AFQMC. Calculating overlaps between an MPS
trial and arbitrary Slater determinants up to a multiplicative error, a crucial
subroutine in MPS-AFQMC, is proven to be #P-hard. Nonetheless, we tested
several promising heuristics in successfully improving fermionic phaseless
AFQMC energies. We also proposed a way to evaluate local energy and force bias
evaluations free of matrix-product operators. This allows for larger basis set
calculations without significant overhead. We showcase the utility of our
approach on one- and two-dimensional hydrogen lattices, even when the MPS trial
itself struggles to obtain high accuracy. Our work offers a new set of tools
that can solve currently challenging electronic structure problems with future
improvements.
