Anyonic Topological Quantum Computation and the Virtual Braid Group

We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation. 1 Classical Trees Louis Kauffman and Sam Lomonaco reconstructed the Fibonacci model of quantum computation in [7] by applying the 2-strand Temperly-Lieb recoupling and the skein relation to braided trees. (The original Fibonacci model is discussed in [10], [5], and [1].) In the context of quantum computation, virtual crossings can be regarded as swap gates. We incorporate virtual crossings into this model as generalized swap gates. This allows us to extend and compute quantum topological invariants to the category of virtual links. In the context of quantum computation, the anyonic model can be used to compute the Jones polynomial of knots and links. Incorporating virtual crossings as generalized swap gates leads to a model that can be used as a quantum algorithm for the Jones polynomial of virtual links. 1 Virtual crossings can be incorporated into the Fibonacci model, but the calculus described in [7] is no longer sufficient to left associate an arbitrarily constructed virtual tree. In this paper, we extend the graphical calculus to virtual braided trees. Recall Artin’s n-strand braid group, Bn. Let {σ1, σ2, . . . σn−1} denote the generators of Bn. The generators of B4, the four strand braid group, are illustrated in figure 1. The n-strand braid group is determined by the Figure 1: Generators of B4 following relations on the generators: • σiσi+1σi = σi+1σiσi+1 • σiσj = σjσi for |i− j| > 1. These relations determine equivalent braids, via the Reidemeister II and III moves, as shown in figure 2. Figure 2: Equivalent braids Extending Bn by the symmetric group results in the n-strand virtual braid group, V Bn [6] [9]. We incorporate virtual crossings by adding the generators v1, v2, . . . vn−1, where vi is a n-strand braid with a single virtual crossing between strand i− 1 and strand i as shown in figure 3. Equivalence classes of virtual braids are determined by the following relations. • v i = Id • vivj = vjvi for |i− j| > 1 2 Figure 3: Virtual generators • vivi+1vi = vi+1vivi+1 • vi+1viσi+1 = σivi+1vi. Diagrammatically, these relations are illustrated in figure 4. Unitary solutions to the Yang-Baxter equation [3] determine unitary representations of the braid group. (Recall that these solutions are universal.) We assume that these representations act on a tensor product; each strand of a braid represents a finite dimensional (usually two dimensional) vector space V . Then in the context of the braid group, order two gates that switch strands lead to the usual definition of the swap gate. That is, in an ndimensional vector space V with basis, {|v1 >, |v2 > . . . |vn >}, an element of V ⊗V is a qudit and a swap gate sends |vi > ⊗|vj > to |vj > ⊗|vi >. This extends the (tensor) representation of the braid group to the virtual braid group. However, we are not working within the context of a tensor representation but instead with a representation that acts on vector spaces associated with trees. The version of the swap gate that we study here is an order two unitary operator. These generalized swap gates also satisfy the relations in the virtual braid group. These gates also have a more complex behavior than the usual definition of a swap gate, a behavior that we will examine later in this paper. Remark 1.1. A tensor representation of the braid group corresponding to V ⊗V ⊗ . . .⊗V , where V is a two dimensional vector space, can be extended with the matrix defined on V ⊗ V :