Exploiting the Hermitian symmetry in tensor network algorithms

TL;DR

This paper demonstrates how exploiting Hermitian symmetry in tensor network algorithms, particularly in PEPS, can significantly reduce computational costs while maintaining physical properties, with practical benchmarks showing up to fourfold speedups.

Contribution

It introduces a method to incorporate Hermitian symmetry into double-layer tensor networks, enabling faster computations and preserving Hermiticity of expectation values.

Findings

Speedup of up to 4x in computation time.

Hermitian symmetry ensures expectation values remain Hermitian.

Benchmark results confirm efficiency gains in CTMRG and HOTRG methods.

Abstract

Exploiting symmetries in tensor network algorithms plays a key role for reducing the computational and memory costs. Here we explain how to incorporate the Hermitian symmetry in double-layer tensor networks, which naturally arise in methods based on projected entangled-pair states (PEPS). For real-valued tensors the Hermitian symmetry defines a $\mathbb{Z}_2$ symmetry on the combined bra and ket auxiliary level of the tensors. By implementing this symmetry, a speedup of the computation time by up to a factor 4 can be achieved, while expectation values of observables and reduced density matrices remain Hermitian by construction. Benchmark results based on the corner transfer matrix renormalization group (CTMRG) and higher-order tensor renormalization group (HOTRG) are presented. We also discuss how to implement the Hermitian symmetry in the complex case, where a similar speedup can be achieved.