Scalable Quantum Monte Carlo Method for Polariton Chemistry via Mixed Block Sparsity and Tensor Hypercontraction Method

We present a reduced-scaling auxiliary-field
quantum
Monte Carlo
(AFQMC) framework designed for large molecular systems and ensembles,
with or without coupling to optical cavities. Our approach leverages
the natural block sparsity of the Cholesky decomposition (CD) of electron
repulsion integrals in molecular ensembles and employs tensor hypercontraction
(THC) to efficiently compress low-rank Cholesky blocks. By representing
the Cholesky vectors in a mixed format, keeping high-rank blocks in
block-sparse form and compressing low-rank blocks with THC, we reduce
the scaling of exchange-energy evaluation from quartic to robust cubic
in the number of molecular orbitals N, while lowering
memory from cubic toward quadratic. Benchmark analyses on one-, two-,
and three-dimensional molecular ensembles (up to ∼1,200 orbitals)
show that (a) the number of nonzeros in Cholesky tensors grows linearly
with system size across dimensions; (b) the average numerical rank
increases sublinearly and does not saturate at these sizes; and (c)
rank heterogeneitysome blocks nearly full rank and many low
rank, naturally motivates the proposed mixed block sparsity and THC
scheme for efficient calculation of exchange energy. We demonstrate
that the mixed scheme yields cubic wall-time scaling with favorable
prefactors and preserves AFQMC accuracy.