Stochastic Resolution-of-the-Identity Auxiliary-Field Quantum Monte Carlo: Scaling Reduction without Overhead

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.