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

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.

v1 - 26 pages + References ; v2, published version - 29 pages + References

References in corpus (6)

• Algebraic Bethe Circuits
• Quantum symmetry algebras of spin systems related to Temperley-Lieb R-matrices
• Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras
• Local invariants of braiding quantum gates -- associated link polynomials and entangling power
• Geometric representations of braid and Yang-Baxter gates
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Cited by in corpus (5)

• Non-hermitian integrable systems from constant non-invertible solutions of the Yang-Baxter equation
• Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains
• Majorana fermions solve the tetrahedron equations as well as higher simplex equations
• Parastatistics in Interacting Periodic Chains Revealed by Peierls Phase Twists and Shifted Conformal Towers
• Almost local integrable models from supersymmetry algebras