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Yang–Baxter equation in all dimensions and universal qudit gates

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Abstract

We construct solutions of the Yang–Baxter equation in any dimension \(d\ge 2\) by directly generalizing the previously found solutions for \(d=2\). We equip those solutions with unitarity and entangling properties. Being unitary, they can be turned into \(2\)-qudit quantum logic gates for qudit-based systems. The entangling property enables each of those solutions, together with all \(1\)-qudit gates, to form a universal set of quantum logic gates.

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Acknowledgments

The author thanks the esteemed referees for their invaluable feedback, which helped to greatly increase the quality and the presentation of this work. The author also thanks Dr. Mansur Saburov for constructive and encouraging discussions. Moreover, I thank Dr. Sinan Kapcak for introducing me to SageMath, which was very useful in generating large matrices and testing the validity of some formulas. Last but not least, I thank Mrs. Elena Ryzhova for helping me by patiently checking specific patterns in large matrices and proofreading the final manuscript, and for fruitful conversations.

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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  1. College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait

    A. Pourkia

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 219, pp. 17–31 https://doi.org/10.4213/tmf10618.

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Pourkia, A. Yang–Baxter equation in all dimensions and universal qudit gates. Theor Math Phys 219, 544–556 (2024). https://doi.org/10.1134/S0040577924040032

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