Cyclic groups and quantum logic gates

Abstract

We present a formula for an infinite number of universal quantum logic gates, which are $4$ by $4$ unitary solutions to the Yang-Baxter (Y-B) equation. We obtain this family from a certain representation of the cyclic group of order $n$. We then show that this {\it discrete} family, parametrized by integers $n$, is in fact, a small sub-class of a larger {\it continuous} family, parametrized by real numbers $\theta$, of universal quantum gates. We discuss the corresponding Yang-Baxterization and related symmetries in the concomitant Hamiltonian.Comment: 12 pages, no figures. Submitted to Physical Review

Quantum teleportation and Birman–Murakami–Wenzl algebra

“…Notes Added. After this paper is done, we are occasionally informed that the Yang-Baxter gates (105) and (108) have been already presented in the preprint [34]. As a matter of fact, these gates are derived in two essentially different approaches.…”

“…As a matter of fact, these gates are derived in two essentially different approaches. We derive such the Yang-Baxter gates in the extended Temperley-Lieb diagrammatical approach [20,21], whereas the authors of [34] obtain them via the algebraic approach of the cyclic group. We study and look for interesting representations of the BMW algebra, which are not involved in [34].…”

“…It is well-known that how to construct an interesting braid representation (2) (or an interesting solution of the Yang-Baxter equation [10]) is always a meaningful and challenge problem in the research frontier, refer to [9,10,11,12,13,21,34], and the same is true for the construction of an interesting representation of the BMW algebra [16], refer to [9,17,18,19,22,23]. In the previous research [21], a method of constructing the Yang-Baxter gate in the extended Temperley-Lieb diagrammatical approach has been proposed, and hence in this section we want to exploit the extended Temperley-Lieb diagrammatical approach to construct interesting representations of the BMW algebra different from the representation using the Temperley-Lieb projector E (8) and the Yang-Baxter gate B (9).…”

A note on "eight-vertex" universal quantum gates

“…On the other hand, they could yield some invariants of links/knots. The studies in this direction have been continued and further expanded by many authors, e.g., in [4,5,6,7,8,9,10,11,12,13].…”

“…All four by four unitary solutions to the Yang-Baxter (Y-B) equation classified in [7], based on [14], are either of the form or similar to a form of (1.1). In [8] we obtained an infinite family of universal quantum logic gates, in the form (1.1), related to cyclic groups of order n. More famously, as a special case, the Bell matrix [4,5,6] is of this particular form. More examples are included but not limited to the ones studied in [1,2,3,4,5,6,7,8,9].…”