Floquetifying stabiliser codes with distance-preserving rewrites

TL;DR

This paper introduces a method to convert stabiliser codes into Floquet codes using distance-preserving rewrites, enabling fault-tolerant implementation with only single- and two-qubit operations while maintaining code properties.

Contribution

It develops a novel Floquetification procedure based on ZX calculus that preserves code distance and logical qubits, applicable to arbitrary stabiliser codes.

Findings

Preserves code distance and logical qubits during transformation.

Requires linear overhead in physical qubits relative to measurement weight.

Enables fault-tolerant implementation with only single- and two-qubit operations.

Abstract

Stabiliser codes with large weight measurements can be challenging to implement fault-tolerantly. To overcome this, we propose a Floquetification procedure which, given a stabiliser code, synthesises a novel Floquet code that only uses single- and two-qubit operations. Moreover, this procedure preserves the distance and number of logicals of the original code. The new Floquet code requires additional physical qubits. This overhead is linear in the weight of the largest measurement of the original code. Our method is based on the ZX calculus, a graphical language for representing and rewriting quantum circuits. However, a problem arises with the use of ZX in the context of rewriting error-correcting codes: ZX rewrites generally do not preserve code distance. Tackling this issue, we define the notion of distance-preserving rewrite that enables the transformation of error-correcting codes without changing their distance. These distance-preserving rewrites are used to decompose arbitrary weight stabiliser measurements into quantum circuits with single- and two-qubit operations. As we only use distance-preserving rewrites, we are guaranteed that a single error in the resulting circuit creates at most a single error on the data qubits. These decompositions enable us to generalise the Floquetification procedure of [arXiv:2307.11136] to arbitrary stabiliser codes, provably preserving the distance and number of logicals of the original code.