Algebraic classification of Hietarinta’s solutions of Yang-Baxter equations: invertible 4 × 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). Includ...
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| Format: | Article |
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2024-12-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP12(2024)067 |
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| author | Somnath Maity Vivek Kumar Singh Pramod Padmanabhan Vladimir Korepin |
| author_facet | Somnath Maity Vivek Kumar Singh Pramod Padmanabhan Vladimir Korepin |
| author_sort | Somnath Maity |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-3718b0fc5d794a34bc53be9a2222d67a |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-3718b0fc5d794a34bc53be9a2222d67a2025-01-05T12:07:04ZengSpringerOpenJournal of High Energy Physics1029-84792024-12-0120241212710.1007/JHEP12(2024)067Algebraic classification of Hietarinta’s solutions of Yang-Baxter equations: invertible 4 × 4 operatorsSomnath Maity0Vivek Kumar Singh1Pramod Padmanabhan2Vladimir Korepin3School of Basic Sciences, Indian Institute of TechnologyCenter for Quantum and Topological Systems (CQTS), NYUAD Research Institute, New York University Abu DhabiSchool of Basic Sciences, Indian Institute of TechnologyC. N. Yang Institute for Theoretical Physics, Stony Brook UniversityAbstract 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.https://doi.org/10.1007/JHEP12(2024)067Lattice Integrable ModelsQuantum GroupsBethe Ansatz |
| spellingShingle | Somnath Maity Vivek Kumar Singh Pramod Padmanabhan Vladimir Korepin Algebraic classification of Hietarinta’s solutions of Yang-Baxter equations: invertible 4 × 4 operators Journal of High Energy Physics Lattice Integrable Models Quantum Groups Bethe Ansatz |
| title | Algebraic classification of Hietarinta’s solutions of Yang-Baxter equations: invertible 4 × 4 operators |
| title_full | Algebraic classification of Hietarinta’s solutions of Yang-Baxter equations: invertible 4 × 4 operators |
| title_fullStr | Algebraic classification of Hietarinta’s solutions of Yang-Baxter equations: invertible 4 × 4 operators |
| title_full_unstemmed | Algebraic classification of Hietarinta’s solutions of Yang-Baxter equations: invertible 4 × 4 operators |
| title_short | Algebraic classification of Hietarinta’s solutions of Yang-Baxter equations: invertible 4 × 4 operators |
| title_sort | algebraic classification of hietarinta s solutions of yang baxter equations invertible 4 4 operators |
| topic | Lattice Integrable Models Quantum Groups Bethe Ansatz |
| url | https://doi.org/10.1007/JHEP12(2024)067 |
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