Exploiting Matrix Symmetries and Physical Symmetries in Matrix Product States and Tensor Trains

TL;DR

This paper discusses how to utilize matrix and physical symmetries in Matrix Product States to reduce computational complexity in simulating quantum many-body systems, providing a unified algebraic framework.

Contribution

It introduces methods to incorporate symmetries into MPS normal forms, enabling more efficient representations of eigenvectors in quantum simulations.

Findings

Symmetries lead to relations between MPS matrices.

Normal forms for symmetric MPS are derived.

Exploiting symmetries reduces degrees of freedom in MPS.

Abstract

We focus on symmetries related to matrices and vectors appearing in the simulation of quantum many-body systems. Spin Hamiltonians have special matrix-symmetry properties such as persymmetry. Furthermore, the systems may exhibit physical symmetries translating into symmetry properties of the eigenvectors of interest. Both types of symmetry can be exploited in sparse representation formats such as Matrix Product States (MPS) for the desired eigenvectors. This paper summarizes symmetries of Hamiltonians for typical physical systems such as the Ising model and lists resulting properties of the related eigenvectors. Based on an overview of Matrix Product States (Tensor Trains or Tensor Chains) and their canonical normal forms we show how symmetry properties of the vector translate into relations between the MPS matrices and, in turn, which symmetry properties result from relations within the MPS matrices. In this context we analyze different kinds of symmetries and derive appropriate normal forms for MPS representing these symmetries. Exploiting such symmetries by using these normal forms will lead to a reduction in the number of degrees of freedom in the MPS matrices. This paper provides a uniform platform for both well-known and new results which are presented from the (multi-)linear algebra point of view.