Computational equivalence of the two inequivalent spinor representations of the braid group in the Ising topological quantum computer

TL;DR

This paper shows that two different spinor representations of the braid group in Ising topological quantum computers are computationally equivalent, meaning they can implement the same set of quantum gates through braiding.

Contribution

It provides explicit matrices and recurrence relations for braidings in both representations, establishing their computational equivalence and proposing a method to identify the representation type in physical systems.

Findings

The two inequivalent representations produce identical sets of quantum gates.

Explicit matrices for braiding operations in both representations are derived.

A process to determine the representation type in physical realizations is proposed.

Abstract

We demonstrate that the two inequivalent spinor representations of the braid group \B_{2n+2}, describing the exchanges of 2n+2 non-Abelian Ising anyons in the Pfaffian topological quantum computer, are equivalent from computational point of view, i.e., the sets of topologically protected quantum gates that could be implemented in both cases by braiding exactly coincide. We give the explicit matrices generating almost all braidings in the spinor representations of the 2n+2 Ising anyons, as well as important recurrence relations. Our detailed analysis allows us to understand better the physical difference between the two inequivalent representations and to propose a process that could determine the type of representation for any concrete physical realization of the Pfaffian quantum computer.