Enhanced Krylov Methods for Molecular Hamiltonians via Tensor Hypercontraction

Abstract: We present a matrix product operator (MPO) construction based on the tensor hypercontraction (THC) format for ab initio molecular Hamiltonians. Such an MPO construction dramatically lowers the memory requirement and cost scaling of Krylov subspace methods. These can find low-lying eigenstates while avoiding local minima and simulate quantum time evolution with 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 matrix product state (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 based on a conventional MPO construction. We provide a detailed theoretical derivation of these statements and conduct supporting numerical experiments to demonstrate the advantage. We also propose that our method is compatible with parallel computing, which is essential for calculating large-size systems.