This paper explores of the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of the Yang-Baxter Equation is a universal gate for quantum computing, in the presence of local unitary transformations. We show that this same R generates a new non-trivial invariant of braids, knots, and links. Other solutions of the Yang-Baxter Equation are also shown to be universal for quantum computation. The paper discusses these results in the context of comparing quantum and topological points of view. In particular, we discuss quantum computation of link invariants, the relationship between quantum entanglement and topological entanglement, and the structure of braiding in a topological quantum field theory.

This paper discusses relationships between topological entanglement and quantum entanglement. Specifically, we propose that it is more fundamental to view topological entanglements such as braids as entanglement operators and to associate to them unitary operators that are capable of creating quantum entanglement.

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten-Reshetikhin-Turaev invariant of three manifolds.

We present a distributed implementation of Shor's quantum factoring algorithm on a distributed quantum network model. This model provides a means for small capacity quantum computers to work together in such a way as to simulate a large capacity quantum computer. In this paper, entanglement is used as a resource for implementing non-local operations between two or more quantum computers. These non-local operations are used to implement a distributed factoring circuit with polynomially many gates. This distributed version of Shor's algorithm requires an additional overhead of O((log N )2 ) communication complexity, where N denotes the integer to be factored.Keywords: Shor's algorithm, factoring algorithm, distributed quantum algorithms, quantum circuit. INTRODUCTIONTo utilize the full power of quantum computation, one needs a scalable quantum computer with a sufficient number of qubits. Unfortunately, the first practical quantum computers are likely to have only small qubit capacity. One way to overcome this difficulty is by using the distributed computing paradigm. By a distributed quantum computer, we mean a network of limited capacity quantum computers connected via classical and quantum channels. Quantum entangled states, in particular generalized GHZ states, provide an effective way of implementing non-local operations, such as, non-local CNOTs and teleportation. 1, 2We use distributed quantum computing techniques to construct a distributed quantum circuit for the Shor factoring algorithm. Let n = log N , where N is the number to be factored. The gate complexity of this particular distributed implementation of Shor's algorithm is O(n 3 ) with O(n 2 ) communication overhead. *In section 2, the general principles of distributed quantum computing are outlined, and two primitive distributed computing operators, cat-entangler and cat-disentangler, are introduced. We use these two primitive operators to implement non-local operations, such as non-local CNOTs and teleportation. Then we discuss how to share the cost of implementing a non-local controlled U , where U can be decomposed into a number of gates. The section ends with an distributed implementation of the Fourier transform.In section 3, we give a detailed description of an implementation of Shor's non-distributed factoring algorithm. This implementation is based on the phase estimation and order finding algorithms. We discuss in detail how to implement "modular exponentiation," which implementation will be used later in this paper as a blueprint for creating a distributed quantum algorithm.In section 4, we implement a distributed factoring algorithm by partitioning the qubits into groups in such a way that each group fits on one of the computers making up the network. We then proceed to replace controlled gates with non-local controlled gates whenever necessary.Further author information: (Send correspondence to Anocha Yimsiriwattana) Anocha Yimsiriwattana: E-mail: ayimsi1@umbc.edu, URL: http:/userpages.umbc.edu/~ayimsi1 Samuel J. Lomonaco Jr.: E-m...

Let K be a CW complex with an aspherical splitting, i.e., with subcomplexes ϋΓ_ and K + such that (a) K-K-UK+ and (b) iΓ_, K 0 =K-ΠK +1 K + are connected and aspherical. The main theorem of this paper gives a practical procedure for computing the homology H*K of the universal cover K of K. It also provides a practical method for computing the algebraic 2-type of K 9 i.e., the triple consisting of the fundamental group π x K, the second homotopy group π 2 K as a TΓilΓ-module, and the first /^-invariant kK.The effectiveness of this procedure is demonstrated by letting K denote the complement of a smooth 2-knot (S 4 , JcS 2 ). Then the above mentioned methods provide a way for computing the algebraic 2-type of 2-knots, thus solving problem 36 of R. H. Fox in his 1962 paper, "Some problems in knot theory." These methods can also be used to compute the algebraic 2-type of 3-manifolds from their Heegaard splittings. This approach can be applied to many other well known classes of spaces. Various examples of the computation are given.