  We explore the use of the stochastic resolution-of-the-identity (sRI) with
the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) method. sRI is
combined with four existing local energy evaluation strategies in ph-AFQMC,
namely (1) the half-rotated electron repulsion integral tensor (HR), (2)
Cholesky decomposition (CD), (3) tensor hypercontraction (THC), or (4) low-rank
factorization (LR). We demonstrate that HR-sRI achieves no scaling reduction,
CD-sRI scales as $\mathcal O(N^3)$, and THC-sRI and LR-sRI scale as $\mathcal
O(N^2)$, albeit with a potentially large prefactor. Furthermore, the
walker-specific extra memory requirement in CD is reduced from $\mathcal
O(N^3)$ to $\mathcal O(N^2)$ with sRI, while sRI-based THC and LR algorithms
lead to a reduction from $\mathcal O(N^2)$ extra memory to $\mathcal O(N)$.
Based on numerical results for one-dimensional hydrogen chains and water
clusters, we demonstrated that, along with the use of a variance reduction
technique, CD-sRI achieves cubic-scaling {\it without overhead}. In particular,
we find for the systems studied the observed scaling of standard CD is
$\mathcal O(N^{3-4})$ while for CD-sRI it is reduced to $\mathcal O(N^{2-3})$.
Once a memory bottleneck is reached, we expect THC-sRI and LR-sRI to be
preferred methods due to their quadratic-scaling memory requirements and their
quadratic-scaling of the local energy evaluation (with a potentially large
prefactor). The theoretical framework developed here should facilitate
large-scale ph-AFQMC applications that were previously difficult or impossible
to carry out with standard computational resources.
