Non-Abelian statistics with mixed-boundary punctures on the toric code

TL;DR

This paper explores how mixed-boundary puncture defects in the toric code can exhibit non-Abelian anyonic behavior, specifically Ising fusion and logical operations, enabling potential quantum information applications.

Contribution

It demonstrates that mixed-boundary punctures in the toric code can produce non-Abelian statistics, expanding the types of defects useful for topological quantum computation.

Findings

Mixed-boundary punctures reproduce Ising fusion rules.

Braiding punctures implements logical Pauli-X operations.

Local lattice defects can exhibit non-Abelian properties.

Abstract

The toric code is a simple and exactly solvable example of topological order realising Abelian anyons. However, it was shown to support non-local lattice defects, namely twists, which exhibit non-Abelian anyonic behaviour [1]. Motivated by this result, we investigated the potential of having non-Abelian statistics from puncture defects on the toric code. We demonstrate that an encoding with mixed-boundary punctures reproduces Ising fusion, and a logical Pauli-$X$ upon their braiding. Our construction paves the way for local lattice defects to exhibit non-Abelian properties that can be employed for quantum information tasks.