Preserving MWPM-Decodability in Fault-Equivalent Rewrites

TL;DR

This paper introduces a set of ZX diagram rewrites that preserve the minimum-weight perfect matching decodability of quantum error correction circuits, enabling fault-tolerant, efficiently decodable syndrome extraction.

Contribution

It formalizes how to maintain MWPM-decodability under ZX rewrites and applies this to construct fault-tolerant syndrome-extraction circuits for matchable codes.

Findings

Rewrites that preserve MWPM-decodability are identified.

Fault-tolerant syndrome extraction circuits are constructed using these rewrites.

The approach enables efficient decoding in fault-tolerant quantum circuits.

Abstract

Decoding a quantum error correction code is generally NP-hard, but corrections must be applied at a high frequency to suppress noise successfully. Matchable codes, like the surface code, exhibit a special structure that makes it possible to efficiently, approximately solve the decoding problem through minimum-weight perfect matching (MWPM). However, this efficiency-enabling property can be lost when constructing implementations for fault-tolerant gadgets such as syndrome-extraction circuits or logical operations. In this work, we take a circuit-centric perspective to formalise how the decoding problem changes when applying ZX rewrites to a ZX diagram with a given detector basis. We demonstrate a set of rewrites that preserve MWPM-decodability of circuits and show that these matchability-preserving rewrites can be used to fault-tolerantly extract quantum circuits from phase-free ZX diagrams. In particular, this allows us to build efficiently decodable, fault-tolerant syndrome-extraction circuits for matchable codes.