Algebraic classification of Hietarinta's solutions of Yang-Baxter equations~:~invertible $4\times 4$ operators

TL;DR

This paper develops algebraic methods to classify invertible solutions of the Yang-Baxter equation for 4x4 operators, extending previous classifications and enabling analysis beyond specific qubit representations.

Contribution

It introduces algebraic ansätze using partition, Clifford, and Temperley-Lieb algebras to classify Yang-Baxter solutions independently of representation.

Findings

Replicates 10 of 11 Hietarinta solution families

Provides a representation-independent classification method

Does not classify the (2,2) Hietarinta class

Abstract

In order to examine the simulation of integrable quantum systems using quantum computers, it is crucial to first classify Yang-Baxter operators. Hietarinta was among the first to classify constant Yang-Baxter solutions for a two-dimensional local Hilbert space (qubit representation). Including the one produced by the permutation operator, he was able to construct eleven families of invertible solutions. These techniques are effective for 4 by 4 solutions, but they become difficult to use for representations with more dimensions. To get over this limitation, we use algebraic ans\"{a}tze to generate the constant Yang-Baxter solutions in a representation independent way. We employ four distinct algebraic structures that, depending on the qubit representation, replicate 10 of the 11 Hietarinta families. Among the techniques are partition algebras, Clifford algebras, Temperley-Lieb algebras, and a collection of commuting operators. Using these techniques, we do not obtain the $(2,2)$ Hietarinta class.