Enhanced Krylov Methods for Molecular Hamiltonians: Reduced Memory Cost and Complexity Scaling via Tensor Hypercontraction

TL;DR

This paper presents a memory-efficient, low-scaling algorithm for applying molecular Hamiltonians to matrix-product states using tensor hypercontraction, improving Krylov methods for quantum simulations.

Contribution

The authors develop a novel tensor hypercontraction-based MPO representation that reduces memory and computational costs in Krylov subspace methods for molecular Hamiltonians.

Findings

Reduced memory cost comparable to bare MPS

Lower computational complexity scaling

Effective for large-scale HPC quantum simulations

Abstract

We introduce an algorithm that is simultaneously memory-efficient and low-scaling for applying ab initio molecular Hamiltonians to matrix-product states (MPS) via the tensor-hypercontraction (THC) format. These gains carry over to Krylov subspace methods, which can find low-lying eigenstates and simulate quantum time evolution while avoiding local minima and maintaining high accuracy. In our approach, the molecular Hamiltonian is represented as a sum of products of four MPOs, each with a bond dimension of only 2. Iteratively applying the MPOs to the current quantum state in MPS form, summing and re-compressing the MPS leads to a scheme with the same asymptotic memory cost as the bare MPS and reduces the computational cost scaling compared to the Krylov method using a conventional MPO construction. We provide a detailed theoretical derivation of these statements and conduct supporting numerical experiments to demonstrate the advantage. Our algorithm is highly parallelizable and thus lends itself to large-scale HPC simulations.