Algebraic classification of Hietarinta's solutions of Yang-Baxter equations~:~invertible $4\times 4$ operators
Somnath Maity, Vivek Kumar Singh, Pramod Padmanabhan, Vladimir Korepin
TL;DR
The paper tackles the problem of classifying constant Yang–Baxter operators for 2D local spaces by developing representation-independent, algebraic constructions. It employs four structures—Clifford algebras, Temperley-Lieb algebras, partition algebras, and commuting operators—to generate YBOs that map, under gauge transformations, to nine of the ten Hietarinta classes (plus the permutation singleton), with the (2,2) class remaining elusive in this framework. The approach yields multiple, often equivalent, realizations of the same Hietarinta classes and provides explicit mappings (via Q) between algebraic generators and the target YBOs. The work advances the systematic production of constant YBOs in higher-dimensional settings, guiding the development of quantum gates and integrable models on quantum hardware. It also uncovers new algebraic YBO solutions that can be benchmarked against known representations and encourages exploration of non-invertible cases and spectral-parameter dependent generalizations.
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 ansätze 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.