Space Reduction in Matrix Product State

TL;DR

This paper introduces a method to reduce the computational complexity of matrix product states by reconstructing them in smaller, density matrix-selected subspaces, enabling efficient simulations of large quantum systems.

Contribution

It presents a novel scheme for space reduction in MPS using density matrices, allowing for efficient and accurate simulations in smaller spaces.

Findings

Reduced space rank decreases rapidly with block size N.

Simulations in reduced spaces reliably extrapolate original results.

Saturated reduced space rank achieves desired accuracy as N approaches infinity.

Abstract

We reconstruct a matrix product state (MPS) in reduced spaces using density matrix. This scheme applies to a MPS built on a blocked quantum lattice. Each block contains $N$ physical sites that have a local space of rank $R$. The simulation in the original spaces of rank $R^N$ is used to construct density matrices for every block. They are diagonalized and only the eigenvectors corresponding to significant diagonal elements are used to transform the original spaces to smaller ones and to reconstruct the MPS in those smaller spaces accordingly. Simulations in the reduced spaces are used to reliably extrapolate the result in unreduced spaces. Moreover, to obtain a required accuracy, the ratio of the reduced space rank over the original decreases quickly with $N$. The reduced space has a saturated rank to obtain a demanded accuracy when $N\rightarrow \infty$.