Floquetifying the Colour Code

TL;DR

This paper introduces a new Floquet code equivalent to the colour code, with simplified measurements and square lattice layout, enhancing fault-tolerance and revealing the relationship between static and dynamic quantum error correction codes.

Contribution

It develops a Floquet code equivalent to the colour code using ZX-calculus, simplifying implementation and connecting static and dynamic code frameworks.

Findings

Floquet code equivalent to the colour code with weight-one or two measurements

Qubits can be laid out on a square lattice

Simplifies fault-tolerant implementation of the colour code

Abstract

Floquet codes are a recently discovered type of quantum error correction code. They can be thought of as generalising stabilizer codes and subsystem codes, by allowing the logical Pauli operators of the code to vary dynamically over time. In this work, we use the ZX-calculus to create new Floquet codes that are in a definable sense equivalent to known stabilizer codes. In particular, we find a Floquet code that is equivalent to the colour code, but has the advantage that all measurements required to implement it are of weight one or two. Notably, the qubits can even be laid out on a square lattice. This circumvents current difficulties with implementing the colour code fault-tolerantly, while preserving its advantages over other well-studied codes, and could furthermore allow one to benefit from extra features exclusive to Floquet codes. On a higher level, as in arXiv:2303.08829, this work shines a light on the relationship between 'static' stabilizer and subsystem codes and 'dynamic' Floquet codes; at first glance the latter seems a significant generalisation of the former, but in the case of the codes that we find here, the difference is essentially just a few basic ZX-diagram deformations.