QSeminar: Introduction to anyonic quantum gates I

Speaker: Claire Levaillant

Title: Introduction to anyonic quantum gates I 

Fibonacci anyons are universal for quantum computation in the sense that any unitary operation of U(d_n) with d_n the number of admissible labelings of the fusion tree obtained from a group of n anyons can be approximated by braiding (exchanging the relative positions of the particles), and this up to desired accuracy. The numbers d_n satisfy the Fibonacci recursion d_(n+2)=d_n+d_(n+1), thus the name given to the quasiparticles.

There is a Hilbert space associated with the basis indexed by all the admissible labelings and the braid group on n strands acts on the this Hilbert space. The latter action is unitary. Universality means that the image of the braid group representation is dense in the unitary group. This result was shown by Freedman-Larsen-Wang (2001).

Each qubit is formed by a group of anyons. In order to be able to perform any operation on any number of qubits, one needs to be able to entangle two of them. When considering two qubits, braiding two anyons from two distinct groups will result in leaving the vector space formed by the two qubits. We say that there is leakage. By the FLW theorem, leakage can be as small as we want. 

In this talk we present a way to entangle two qubits exactly, without leakage, by introducing some measurements operations.