QSpace - An open-source tensor library for Abelian and non-Abelian symmetries

TL;DR

QSpace is an open-source tensor library that efficiently exploits a wide range of abelian and non-abelian symmetries in tensor network states for quantum many-body systems, simplifying symmetry handling in algorithms.

Contribution

It introduces a versatile, bottom-up C++ library that explicitly computes Clebsch-Gordan tensors for all classical Lie algebra symmetries, enabling seamless integration into tensor network algorithms.

Findings

Supports arbitrary combinations of abelian and non-abelian symmetries.

Provides a user-friendly interface hiding symmetry complexities.

Open source implementation available for community use.

Abstract

This is the documentation for the tensor library QSpace (v4.0), a toolbox to exploit `quantum symmetry spaces' in tensor network states in the quantum many-body context. QSpace permits arbitrary combinations of symmetries including the abelian symmetries $\mathbb{Z}_n$ and $U(1)$, as well as all non-abelian symmetries based on the semisimple classical Lie algebras: $A_n$, $B_n$, $C_n$, and $D_n$, or respectively, the special unitary group SU($n$), the odd orthogonal group SO($2n+1$), the symplectic group Sp($2n$), and the even orthogonal group SO($2n$). The code (C++ embedded via the MEX interface into Matlab) is available open source as of QSpace v4.0 at https://bitbucket.org/qspace4u/ under the Apache 2.0 license. QSpace is designed as a bottom-up approach for non-abelian symmetries. It starts from the defining representation and the respective Lie algebra. By explicitly computing and tabulating generalized Clebsch-Gordan coefficient tensors, QSpace is versatile in the type of operations that it can perform across all symmetries. At the level of an application, much of the symmetry-related details are hidden within the QSpace C++ core libraries. Hence when developing tensor network algorithms with QSpace, these can be coded (nearly) as if there are no symmetries at all, despite being able to fully exploit general non-abelian symmetries.