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A note on eight-vertex universal quantum gates and a family of non-eight-vertex gates
Abstract
Kauffman and Lomonaco initiated the idea that unitary solutions to Yang-Baxter equation are crucial tools in studying the relationship between quantum entanglement and topological entanglement. Hence we believe creating a more explicit formalism and more examples of those solutions is an important contribution to this field of study. Many well-known and well-studied four-by-four universal quantum logic gates in the literature are of a specific form, the so called eight-vertex form. In the first part of this paper, we provide explicit formulas in terms of matrix entries, which are the necessary and sufficient conditions for a four-by-four complex matrix, of the eight-vertex form, to be a unitary solution to Yang-Baxter equation. Hence, considering the conditions needed for
being entangling, we give the explicit description of universal quantum logic gates of such a specific form, in terms of matrix entries. In the second part of this paper, we create a family of examples of unitary solutions to Yang-Baxter equation of non-eight-vertex form, using a hint from the cyclic group.
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