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

TL;DR

This work introduces 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 and lends itself to large-scale HPC 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.

Figures (12)

• Figure 1: Graphical representation of the THC factorization to approximate the Coulomb (electron repulsion integral) tensor.
• Figure 2: Graphical tensor network representation of an MPS and MPO. The logical wavefunction and operator are obtained by contracting the matrices $A[i]^{n_i}$ and $W[i]^{m_i n_i}$, respectively.
• Figure 3: Multiplying an MPO with an MPS and subsequent compression. We first contract the tensors along the physical axis. Then, the MPS is transformed into right-canonical form by QR decompositions. Next, we employ SVDs from left to right to reduce the bond dimension by discarding the smallest singular values and merging $S$ and $V^{\dagger}$ matrices into the next site.
• Figure 4: $G_{\mu \sigma, \nu \sigma^{\prime}}$ in Eq. \ref{['eq:G_mu_nu_def']} is composed of four layers of MPOs (square tensors) with bond dimension $2$. As specified by the arrows, we contract and compress the layers one at a time with the MPS (orange tensors).
• Figure 5: Convergence of the water molecule's ground and first excited state calculation using the Lanczos algorithm based on our THC-MPO. We restart the iteration at the 45th step for the first excited state finding.