Topological-Like Features in Diagrammatical Quantum Circuits

Abstract

In this paper, we revisit topological-like features in the extended Temperley-Lieb diagrammatical representation for quantum circuits including the teleportation, dense coding and entanglement swapping. We perform these quantum circuits and derive characteristic equations for them with the help of topological-like operations. Furthermore, we comment on known diagrammatical approaches to quantum information phenomena from the perspectives of both tensor categories and topological quantum field theories. Moreover, we remark on the proposal for categorical quantum physics and information to be described by dagger ribbon categories. -23, 2006), in which he has been proposing categorical quantum physics & information to be described by dagger ribbon categories and emphasizing the functor between Abramsky and Coecke's categorical quantum mechanics and his extended Temperley-Lieb categorical approach to be the same type as those defining topological quantum field theories. As a theoretical physicist, however, the proposer himself has to admit that these arguments are rather mathematical type so that they are hardly appreciated by physicists because physics is such a great field including various kinds of topics and topology only plays important roles in a limited number of physical problems in the present knowledge. On the other hand, this proposal is suggesting either that fundamental objects in the physical world are string-like (even brane-like) and satisfy the braid statistics or that quasi-particles of many-body systems (or fundamental particles at the Planck energy scale) obey the braid statistics and have an effective (or a new internal) degree of freedom called the "twist spin", so that the braiding and twist operations for defining ribbon categories would obtain a reasonable and physical interpretation. Furthermore, this name "categorical quantum physics and information" hereby refers to quantum physics and information which can be recast in terms of the language of categories, and it is a simple and intuitional generalization of the name "categorical quantum mechanics" because the latter does not recognize conformal field theories, topological quantum field theories, quantum gravity and string theories which have been already described in the categorical framework by different research groups. Moreover, the proposal categorical quantum physics and information has been strongly motivated by the present study in quantum information phenomena and theory, and it is aimed at setting up a theoretical platform on which both categorical quantum mechanics and topological quantum computing by Freedman, Larsen and Wang are allowed to stand.

Braid group, Temperley-Lieb algebra, and quantum information and computation

“…As a remark, the above examples will be exploited in the following sections as topological-like diagrammatical tricks [34] in the extended TL diagrammatical configuration for quantum information and computation.…”

“…where lower indices A, B, C are neglected and which is called the characteristic equation for quantum teleportation in our previous work [34]. It catches the aim of a successful teleportation: Charlie performs quantum measurement in his system as he does in Bob's system, though they are far away from each other.…”

“…It has two diagrammatical terms and each one consists of two teleportation processes for sending a two-qubit. The swap gate P , denoted by (34), is an element of the extended TL category, i.e., Figure 25 which has four diagrammatical terms and each one represents an algebraic term in (34). The CNOT gate, a linear combination of products of Pauli matrices,…”

“…As a remark, the extended TL category is a low-dimensional "topological" model for universal quantum computation, and "topological" or topological-like features [34] are expected to be helpful to look for new quantum algorithms. Figure 26: CNOT gate in the extended TL category.…”

“…In a joint article with Kauffman and Werner [31], we make a survey of diagrammatical tensor calculus and matrix representations, and explore topological and algebraic structures underlying multipartite entanglements. In this article as a further extension of our work [32,33,34], we describe quantum information and computation (especially quantum teleportation [8,9]) in the language of the braid group and Temperley-Lieb (TL) algebra [35]. We propose the braid teleportation, teleportation swapping and virtual braid teleportation, and devise the extended TL diagrammatical rules to describe quantum teleportation, entanglement swapping, universal quantum computation and quantum information flow.…”