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

TL;DR

This work addresses the computational bottleneck of exchange-energy evaluation in AFQMC when applied to polaritonic chemistry and large molecular ensembles by introducing a mixed block-sparsity and tensor hypercontraction (BS-THC) representation of Cholesky tensors. By exploiting block sparsity from spatial locality and the low-rank nature of most Cholesky blocks, the authors route high-rank blocks to a block-sparse path and compress genuinely low-rank blocks with THC, achieving an overall $O(N^3)$ scaling and memory near $O(N^2)$ while preserving AFQMC accuracy. Numerical benchmarks on 1D, 2D, and 3D molecular ensembles (up to ~1200 orbitals) reveal linear nonzero growth per block and sublinear average rank growth, along with pronounced rank heterogeneity that motivates the mixed BS-THC strategy. The results enable scalable, predictive AFQMC simulations of cavity-modified chemistry and strongly correlated polaritonic matter, with potential extensions to other ERI-dominated methods such as CCSD. This mixed approach thus provides a practical route to cubic-scaling exchange-energy evaluation in large quantum ensembles.

Abstract

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 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, 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 motivating the proposed mixed block sparsity and THC scheme for efficient calculation of exchange energy. We demonstrate that the mixed scheme yields cubic CPU-time scaling with favorable prefactors and preserves AFQMC accuracy.

Figures (4)

• Figure 1: Schematic 1D (top), 2D (bottom left), and 3D (bottom right) ensembles used in benchmarks. Random molecular orientations are used to avoid any symmetries.
• Figure 2: Sparsity of the averaged Cholesky tensor $\langle L\rangle$ for a) 1D (120 molecules), b) 2D ($11^2=121$ molecules), and c) 3D ($5^3=125$ molecules) ensembles: block diagonality with a narrow neighbor stencil is evident in all cases. d) Average NNZ per $L^\gamma$ versus $N$; the linear trend holds across dimensions with geometry-dependent slopes, supporting $\mathrm{NNZ}(L^\gamma)=\mathcal{O}(N)$.
• Figure 3: (a) Average numerical rank $\bar{R}$ of $L^\gamma$ versus $N$ for 1D/2D/3D ensembles at fixed tolerance: $\bar{R}$ grows sublinearly and does not saturate up to $\sim$1200 orbitals, implying super-cubic/sub-quartic scaling for pure THC. (b) Rank $R_\gamma$ versus Cholesky index: many genuinely low rank vectors coexist with a subset that is near full rank, motivating the split of Cholesky tensors into BS and THC sets in Eq. \ref{['eq:decision_rule']}. $\tau_{\mathrm{CD}} = \tau_{\mathrm{THC}}=10^{-4}$ is used.
• Figure 4: CPU scaling and accuracy of exchange-energy evaluation for pure CD (quartic, black line), pure THC (super-cubic/sub-quartic due to rank growth, blue line), and the proposed mixed BS-THC method (robust cubic, red line). The fitted scaling is included in the legend. Overall, the mixed scheme consistently achieves $\sim\mathcal{O}(N^3)$ scaling while preserving accuracy.