  Phaseless Auxiliary-Field Quantum Monte Carlo (ph-AFQMC) has recently emerged
as a promising method for the production of benchmark-level simulations of
medium to large-sized molecules, due to its accuracy and favorable polynomial
scaling with system size. Unfortunately the memory footprint of standard energy
evaluation algorithms are non-trivial, which can significantly impact timings
on graphical processing units (GPUs) where memory is limited. Previous attempts
to reduce scaling by taking advantage of the low rank structure of the
Coulombic integrals have been successful, but are significantly limited by high
prefactors, rendering the utility limited to very large systems. Here, we
present a complementary, cubic scaling route to reduce memory and computational
scaling based on the low rank of the Coulombic interactions between localized
orbitals, focusing on the application to phaseless AFQMC. We show that the
error due to this approximation, which we term Localized Orbital AFQMC
(LO-AFQMC), is systematic and controllable via a single variable, and is
computationally favorable even for small systems. We present results
demonstrating a robust retention of accuracy versus both experiment and full
ph-AFQMC for a variety of test cases chosen for their potential difficulty for
localized orbital based methods, including the singlet-triplet gaps of
polyacenes benzene through pentacene, the heats of formation for a set of
platonic hydrocarbon cages, and the total energy of ferrocene (Fe(Cp)$_2$).
Finally, we reproduce our previous result of the gas phase ionization energy of
Ni(Cp)$_2$, agreeing with full ph-AFQMC to within statistical error while using
less than a fifteenth of the computer time.
