Optimal Symbolic Construction of Matrix Product Operators and Tree Tensor Network Operators

TL;DR

This paper presents an enhanced method for constructing matrix product and tree tensor network operators, improving efficiency especially for Hamiltonians with shared terms, with applications demonstrated in quantum systems modeling.

Contribution

It introduces a symbolic Gaussian elimination preprocessing step combined with bipartite-graph methods, advancing tensor network construction techniques for quantum simulations.

Findings

Improved tensor network construction efficiency.

Sub-linear growth of bond dimension in a quantum system model.

Benchmarking shows performance gains over previous algorithms.

Abstract

This research introduces an improved framework for constructing matrix product operators (MPOs) and tree tensor network operators (TTNOs), crucial tools in quantum simulations. A given (Hamiltonian) operator typically has a known symbolic "sum of operator strings" form that can be translated into a tensor network structure. Combining the existing bipartite-graph-based approach and a newly introduced symbolic Gaussian elimination preprocessing step, our proposed method improves upon earlier algorithms in cases when Hamiltonian terms share the same prefactors. We test the performance of our method against established ones for benchmarking purposes. Finally, we apply our methodology to the model of a cavity filled with molecules in a solvent. This open quantum system is cast into the hierarchical equation of motion (HEOM) setting to obtain an effective Hamiltonian. Construction of the corresponding TTNO demonstrates a sub-linear increase of the maximum bond dimension.