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

## TL;DR

This paper introduces a new algorithm that reduces memory and computational costs when simulating molecular systems using tensor hypercontraction and Krylov subspace methods.

## Contribution

The novel contribution is a memory-efficient and low-scaling algorithm for ab initio molecular Hamiltonians using tensor-hypercontraction and matrix-product states.

## Key findings

- The algorithm achieves the same memory cost as the bare MPS while reducing computational cost scaling.
- Numerical experiments confirm the theoretical advantages of the proposed method.
- The method is highly parallelizable, making it suitable for large-scale high-performance computing 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 recompressing 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.