A Hierarchy of Anyon Models Realised by Twists in Stacked Surface Codes

TL;DR

This paper explores how twists in stacked surface codes can be used to realize a hierarchy of anyon models, enabling fault-tolerant logical operations like S and CZ gates through braiding.

Contribution

It identifies conditions for twists to behave as anyons and demonstrates how these can implement universal logical gates in stacked surface codes.

Findings

Twists correspond to a hierarchy of anyon models generalizing Ising anyons.

Braiding twists implements S and CZ gates.

Universal gate set generated by H and CNOT gates in stacked codes.

Abstract

Braiding defects in topological stabiliser codes can be used to fault-tolerantly implement logical operations. Twists are defects corresponding to the end-points of domain walls and are associated with symmetries of the anyon model of the code. We consider twists in multiple copies of the 2d surface code and identify necessary and sufficient conditions for considering these twists as anyons: namely that they must be self-inverse and that all charges which can be localised by the twist must be invariant under its associated symmetry. If both of these conditions are satisfied the twist and its set of localisable anyonic charges reproduce the behaviour of an anyonic model belonging to a hierarchy which generalises the Ising anyons. We show that the braiding of these twists results in either (tensor products of) the S gate or (tensor products of) the CZ gate. We also show that for any number of copies of the 2d surface code the application of H gates within a copy and CNOT gates between copies is sufficient to generate all possible twists.