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.
@misc{schweikart2026preservingmwpmdecodabilityfaultequivalentrewrites, title={Preserving MWPM-Decodability in Fault-Equivalent Rewrites}, author={Maximilian Schweikart and Linnea Grans-Samuelsson and Aleks Kissinger and Benjamin Rodatz}, year={2026}, eprint={2603.19522}, archivePrefix={arXiv}, primaryClass={quant-ph}, url={https://arxiv.org/abs/2603.19522}, }