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Theoretical and Mathematical Physics

Yang–Baxter equation in all dimensions and universal qudit gates

Arash Pourkia

Published Apr 1, 2024• 4 citations
Publisher

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.

DOI
10.1134/s0040577924040032

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We construct solutions of the Yang–Baxter equation in any dimension d≥2d\ge 2d≥2 by directly generalizing the previously found solutions for d=2d=2d=2 . We equip those solutions with unitarity and entangling properties. Being unitary, they can be turned into 222 -qudit quantum logic gates for qudit-based systems. The entangling property enables each of those solutions, together with all 111 -qudit gates, to form a universal set of quantum logic gates.
We present a new class of solutions to the parameter-dependent Yang–Baxter equation across all dimensions, which includes a significant subclass of unitary and entangling solutions. In any dimension d ≥ 2 , we construct a well-structured matrix R ( u ) that exhibits intriguing and useful symmetries. A key feature of R ( u ) is its composition from four monomial-based layers: R ( u ) = R a ( u ) + R b ( u ) + R x ( u ) + R y ( u ) , where a ( u ) , b ( u ) , x ( u ) and y ( u ) are four sets of monomial complex functions of the form, a u n a , b u n b , x u n x and y u n y , respectively. This well-designed structure plays a crucial role in demonstrating that R ( u ) is indeed a solution to the Yang–Baxter equation and in establishing the conditions for its unitarity and entangling properties. Additionally, it allows us to identify interesting non-trivial subfamilies with these properties. As is widely recognized, unitary and entangling solutions to the Yang–Baxter equation serve as universal quantum logic gates for qudit quantum computing . Moreover, the search for new solutions to the Yang–Baxter equation in higher dimensions is a common endeavor in both mathematics and physics .
The Yang-Baxter equation and it's various forms have applications in many fields, including statistical mechanics, knot theory, and quantum information. Unitary solutions of the braided Yang-Baxter equation are of particular interest as quantum gates for topological quantum computers. We demonstrate a simple construction for solutions in any dimension, which are both unitary and universal for quantum computation. We also fully classify a family of solutions to certain generalized Yang-Baxter equations and prove that certain instances of the equation only have solutions that are scalar multiples of the identity.
Unitary solutions to the Yang-Baxter equation are important to quantum information science because they lead to unitary representations of the braid group, which can be used to design quantum logic gates that make up topological quantum circuits. By finding new unitary solutions to the Generalized Yang-Baxter equation in low dimensions and classifying them, we will be able to find new representations of the braid group which may lead to new designs for quantum logic gates used in quantum computers. Because it is extremely difficult to find solutions to the Generalized Yang-Baxter equation, we will narrow our search to set-theoretical solutions, that is, solutions that are also permutation matrices.
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 \eqref{8vertexform} \cite{kaufman etal 05-1,kaufman etal 05-2}, or {\it similar} to it. We present a formalism for universal quantum logic gates of such a form. First, we provide explicit formulas in terms of matrix entries, which are the necessary and sufficient conditions for such a matrix to be a solution to the Yang-Baxter equation \eqref{byb}. Then, combining this with the conditions needed for being unitary \eqref{unitarycond} and being entangling \eqref{entanglingcond}, we give a full description of entangling unitary solutions to the Yang-Baxter equation (hence, universal quantum logic gates) of such a specific form. We investigate in detail all the possible cases where some of the eight main entries might or might not be zero.
We introduce an infinite family of universal quantum logic gates that includes not only higher-dimensional versions of the usual SWAP and i\mathbf{i}iSWAP gates, but also their previously known extensions. This family consists of permutation-like matrices with nonzero entries of the form eiαie^{\mathbf{i}\alpha_i}eiαi​, where the αi\alpha_iαi​ are arbitrary real numbers. Moreover, we show that these gates, which we refer to as αSWAP\alpha SWAPαSWAP, provide unitary solutions to the constant Yang--Baxter equation in all dimensions.