Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang–Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the PPT algebra, which guides the construction of multipartite symmetric states. The virtual knot theory, having permutation as a virtual crossing, provides a topological language describing quantum computation as having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley–Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang–Baxter equations; and virtual Temperley–Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the ABPK diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies non-trivial unitary braid representations with universal quantum gates, derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.
Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang–Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the PPT algebra, which guides the construction of multipartite symmetric states. The virtual knot theory, having permutation as a virtual crossing, provides a topological language describing quantum computation as having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley–Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang–Baxter equations; and virtual Temperley–Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the ABPK diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies non-trivial unitary braid representations with universal quantum gates, derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.
. This paper is an introduction to relationships between quantum topology and quantum computing. We show how knots are related not just to braiding and quantum operators, but to quantum set theoretical foundations, algebras of fermions, and we show how the operation of negation in logic, seen as both a value and an operator, can generate the fusion algebra for a Majorana fermion. We call negation in this mode the mark, as it operates on itself to change from marked to unmarked states. The mark viewed recursively as a simplest discrete dynamical system naturally generates the fermion algebra, the quaternions and the braid group representations related to Majorana fermions. The paper begins with these fundamentals. It then discusses unitary solutions to the Yang-Baxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and gives an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and applies these methods to produce unitary representations of the braid groups that are dense in the unitary groups. These methods are rooted in the bracket state sum model for the Jones polynomial. A self-contained study of the quantum universal Fibonacci model is given. Results are applied to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and theWitten-Reshetikhin-Turaev invariant of three manifolds. Two constructions are given for the Fibonacci model, one based in Temperley-Lieb recoupling theory, the other quite elementary and also based on the Temperley-Lieb algebra. This paper begins an exploration of quantum epistemology in relation to the structure of discrimination as the underpinning of basic logic, perception and measurement. Introduction. This paper is an introduction to relationships between quantum topology and quantum computing. We take a foundational approach, showing how knots are related not just to braiding and quantum operators, but to quantum set theoretical foundations and algebras of fermions. We show how the operation of negation in logic, seen as both a value and an operator, can generate the fusion algebra for a Majorana fermion, a particle that is its own anti-particle and interacts with itself either to annihilate itself or to produce itself. We call negation in this mode the mark , as it operates on itself to change from marked to unmarked states.
In this paper, we explore algebraic structures and low dimensional topology underlying quantum information and computation. We revisit quantum teleportation from the perspective of the braid group, the symmetric group and the virtual braid group, and propose the braid teleportation, the teleportation swapping and the virtual braid teleportation, respectively. Besides, we present a physical interpretation for the braid teleportation and explain it as a sort of crossed measurement. On the other hand, we propose the extended Temperley--Lieb diagrammatical approach to various topics including quantum teleportation, entanglement swapping, universal quantum computation, quantum information flow, and etc. The extended Temperley--Lieb diagrammatical rules are devised to present a diagrammatical representation for the extended Temperley--Lieb category which is the collection of all the Temperley--Lieb algebras with local unitary transformations. In this approach, various descriptions of quantum teleportation are unified in a diagrammatical sense, universal quantum computation is performed with the help of topological-like features, and quantum information flow is recast in a correct formulation. In other words, we propose the extended Temperley--Lieb category as a mathematical framework to describe quantum information and computation involving maximally entangled states and local unitary transformations.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly , from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly from separable basis states important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.
It has been conjectured that quantum entanglement operators can be lifted to braiding operators by the way of the topological quantum field theory axioms set forth by Witten and Atiyah. Moreover, it can be readily shown that quantum link invariants need entanglement to construct topological invariants. Given these results and the already dense mathematical framework underlying topology and quantum field theory, we propose that, through the usage of quantum algebra and bracket models, we can identify a significant area of overlap where entangling R-matrix solutions to the Yang-Baxter equation can be used to construct invariants of knots and links. Such a connection illuminates the relationship between quantum and topological entanglement, and allows us to draw novel insights about properties of entangling gates, for example the question of what it is that allows some entangling R-matrices to generate topological invariants when others do not. In conclusion, by studying the boundary between topological and quantum entanglement we construct novel topological invariants that can have significant impact on the study of quantum computing and can help make progress towards the problem of representing the Artin braid group as unitary matrices.