Implementation of Clifford gates in the Ising-anyon topological quantum computer

TL;DR

This paper proves that in Ising topological quantum computers, braiding of anyons can implement all single-qubit Clifford gates and two-qubit gates, but not all multi-qubit Clifford gates, highlighting the limitations of braiding-based quantum computation.

Contribution

It provides a general proof of the realizability of Clifford gates via braiding in Ising anyons and characterizes the scope and limitations of such implementations.

Findings

All single-qubit Clifford gates can be implemented by braiding.

Two-qubit braiding gates generate the entire two-qubit Clifford group.

Not all multi-qubit Clifford gates can be realized by braiding in this scheme.

Abstract

We give a general proof for the existence and realizability of Clifford gates in the Ising topological quantum computer. We show that all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates. We find that the braiding gates for two qubits exhaust the entire two-qubit Clifford group. Analyzing the structure of the Clifford group for n \geq 3 qubits we prove that the the image of the braid group is a non-trivial subgroup of the Clifford group so that not all Clifford gates could be implemented by braiding in the Ising topological quantum computation scheme. We also point out which Clifford gates cannot in general be realized by braiding.