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European Journal of Pure and Applied Mathematics

Generalized SWAP and iSWAP and Solutions to the Yang-Baxter Equation in All Dimensions

Arash Pourkia

Published Aug 1, 2025• 1 citation
Publisher

Abstract

We introduce an infinite family of universal quantum logic gates that includes not only higher-dimensional versions of the usual SWAP and $\mathbf{i}$SWAP gates, but also their previously known extensions. This family consists of permutation-like matrices with nonzero entries of the form $e^{\mathbf{i}\alpha_i}$, where the $\alpha_i$ are arbitrary real numbers. Moreover, we show that these gates, which we refer to as $\alpha SWAP$, provide unitary solutions to the constant Yang--Baxter equation in all dimensions.

DOI
10.29020/nybg.ejpam.v18i3.6344

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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.
We present explicit descriptions for the swap gate and the iswap gate in any arbitrary dimension d≥2d \geq 2d≥2, in terms of permutation matrices. Moreover, we unify these gates by introducing a more general gate xSWAP which includes SWAP and iSWAP for x=1x=1x=1 and x=ix=ix=i (i.e. −1\sqrt{-1}−1​), respectively. The higher dimensional xSWAP e.g., the swap and iswap gates for d>2d > 2d>2 serve as quantum logic gates that operate on two ddd-level qudits. For d=2d=2d=2, it is well known that iSWAP unlike SWAP is universal for quantum computing. We will prove this fact for xSWAP in any dimension ddd, when x≠±1x \neq \pm 1x=±1. Our explicit representation of xSWAP by a permutation matrix facilitates the proof, greatly.
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 will present solutions to the constant Yang-Baxter equation, in any dimension nnn. More precisely, for any nnn, we will create an infinite family of n2n^2n2 by n2n^2n2 matrices which are solutions to the constant Yang-Baxter equation. The total number of non-vanishing entries of such a matrix is 4n24n^24n2 for nnn even, and 4(n−1)(n)+14(n-1)(n)+14(n−1)(n)+1 for nnn odd. We will also present the unitary conditions for those matrices. Moreover, we discuss the entangling property of those matrices.
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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.
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 .