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International journal of mathematics and computation

A note on eight-vertex universal quantum gates and a family of non-eight-vertex gates

Arash Pourkia

Published Apr 17, 2020• 0 citations
OpenAlex

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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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.
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 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.
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.
Using the Kauffman-Lomonaco method, some two-qutrits universal quantum gates are derived from the nine-dimensional unitary solutions of the Yang-Baxter equations associated with algebraic structures like the partial transpose operator and the dihedral group, which admit three dimensional representations. The Yang-Baxterization method given by Zhang-Kauffman-Ge is continuously used to obtain two-qutrits quantum gates and certain Hamiltonians for the evolution of the quantum gates are obtained, being such Hamiltonians interpreted as physical Hamiltonians of chain of particles of spin 1. Finally, the generalization for systems of two qudits is presented in the case of Yang-Baxterization of representations of braided monoidal algebra like the BH algebra and the bicolored Birman-Wenzl-Muraki algebra. For these algebras the corresponding two-qudits quantum gates are constructed jointly with the associated Hamiltonians interpreted like physical chains of particles with spin d. It is conjectured that the derived two-qdits quantum gates and the Hamiltonians may be implemented over bi-dimensional lattice systems like anyons systems or more generally over any physical systems ruled by the Yang-Baxter equations.
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.