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arXiv:2603.19522

Preserving MWPM-Decodability in Fault-Equivalent Rewrites

Maximilian Schweikart, Linnea Grans-Samuelsson, Aleks Kissinger, Benjamin Rodatz

quant-phSubmitted Mar 19, 2026• Announced Mar 23, 2026• 0 citations
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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.

Comments
23 pages. Submitted to QPL 2026

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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.
Efficient and realistic error decoding is crucial for fault-tolerant quantum computation (FTQC) on near-term devices. While decoding is a classical post-processing task, its effectiveness depends on accurately modeling quantum noise, which is hardware-dependent. In particular, correlated bit-flip (XXX) and phase-flip (ZZZ) errors often arise under circuit-level noise. We introduce the Iterative Reweighting Minimum-Weight Perfect Matching (IRMWPM) decoder, which systematically incorporates such correlations to enhance quantum error correction. Our method leverages fault-detection patterns to guide reweighting: correlated XXX and ZZZ detection events are identified, and their conditional probabilities update weights on the primal and dual lattices. This iterative procedure improves handling of realistic error propagation in a hardware-agnostic yet noise-aware manner. We prove that IRMWPM converges in finite time while preserving the distance guarantee of MWPM. Numerical results under circuit-level noise show substantial improvements. For distances ≥17\geq 17≥17 and physical error rates ≤0.001\leq 0.001≤0.001, IRMWPM reduces logical error rates by over 20x with only a few iterations. It also raises the accuracy threshold from 1% to 1.16%, making it practical for near-term real-time decoding. Extrapolated estimates suggest that to reach logical error rate 10−1610^{-16}10−16, IRMWPM requires distance d=31d=31d=31, while standard MWPM needs d=50d=50d=50, implying a major reduction in qubit overhead.
In this work, we introduce a fast implementation of the minimum-weight perfect matching (MWPM) decoder, the most widely used decoder for several important families of quantum error correcting codes, including surface codes. Our algorithm, which we call sparse blossom, is a variant of the blossom algorithm which directly solves the decoding problem relevant to quantum error correction. Sparse blossom avoids the need for all-to-all Dijkstra searches, common amongst MWPM decoder implementations. For 0.1% circuit-level depolarising noise, sparse blossom processes syndrome data in both X and Z bases of distance-17 surface code circuits in less than one microsecond per round of syndrome extraction on a single core, which matches the rate at which syndrome data is generated by superconducting quantum computers. Our implementation is open-source, and has been released in version 2 of the PyMatching library.
Sparse-Blossom Decoding in o(1)o(1)o(1) Time
Matching-based decoding is widely used in quantum error correction, and accelerating it is key to enabling fast and scalable fault-tolerant quantum computation. Minimum-weight perfect matching (MWPM) decoding provides rigorous guarantees for error suppression, while sparse blossom enables its practical implementation at modest problem sizes. However, the runtime of existing sparse-blossom implementations unavoidably increases with problem size, motivating a rigorous parallelization framework that guarantees correctness and a runtime shorter than the syndrome-extraction timescale. Here, we present such a framework and prove that the resulting parallel sparse-blossom algorithm produces the same correction as the original, non-parallel sparse blossom. For the rotated surface code with code distance ddd and physical error rates below a finite threshold, we prove that the average parallel runtime of decoding for O(d)O(d)O(d) rounds of syndrome extraction is upper bounded by a quasi-polylogarithmic function of ddd. For a ddd-round decoding window, this implies that the average parallel runtime per round is o(1)o(1)o(1). We also perform numerical simulation to identify conditions under which the parallel runtime per round decreases with increasing code distance. These results suggest that increasing code distance need not lead to longer parallel decoding times, providing a foundation for scalable parallel matching-based decoding.
Two circuits are considered to be equivalent under noise if the effect of faults on one circuit is no worse than the effect of faults on the other circuit. We call this relationship fault equivalence. Fault equivalence offers a way to transform circuits while provably preserving their fault-tolerant properties, enabling a framework for fault-tolerant circuit synthesis and optimisation that is correct by construction. The ZX calculus offers a diagrammatic way to represent and reason about quantum circuits and is a useful tool for manipulating circuits while preserving fault equivalence. For this, the usual set of ZX rewrites has to be restricted to not only preserve the underlying linear map represented by the diagram, but also fault equivalence. In this work, we provide a set of ZX rewrites that are sound and complete for fault equivalence of Clifford ZX diagrams. This means that any equivalence that can be derived using the proposed rules is certain to be correct, and any correct equivalence can be derived using only these rules. For this, we utilise diagrammatic constructions called fault gadgets to reason about arbitrary, possibly correlated Pauli faults in ZX diagrams. Fault gadgets allow us to separate the diagram into a fault-free part, which captures the noise-free behaviour of a diagram, and a noisy part that enumerates the effects of all possible faults. Using this, we provide a unique normal form for ZX diagrams under noise and show that any diagram can be brought into this normal form using our proposed rule set. Leveraging the constructive nature of our proof of completeness, we implemented a software library to check whether two ZX diagrams with given noise models are fault-equivalent.
The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process---it is the algorithm which computes the operations needed to correct or compensate for the errors according to the measured syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes. In this work, we develop a fault-tolerant decoder for the surface code, capable of efficient operation for qubits and qudits of any dimension, generalizing the decoder first introduced by Bravyi and Haah [Phys. Rev. Lett. 111, 200501 (2013)]. We study its performance when both the physical qudits and the syndromes measurements are subject to generalized uncorrelated bit-flip noise (and the higher-dimensional equivalent). We show that, with appropriate enhancements to the decoder and a high enough qudit dimension, a threshold at an error rate of more than 88%8 can be achieved.
A key challenge in fault-tolerant quantum computing is synthesising and optimising circuits in a noisy environment, as traditional techniques often fail to account for the effect of noise on circuits. In this work, we propose and numerically verify a framework for designing fault-tolerant quantum circuits that are correct by construction. The framework starts with idealised specifications of fault-tolerant gadgets and refines them using provably sound basic transformations. To reason about manipulating circuits while preserving their error correction properties, we define fault equivalence; two circuits are considered fault-equivalent if all undetectable faults on one circuit have a corresponding fault on the other. This guarantees that the effect of undetectable faults on both circuits is the same. We argue that fault equivalence is a concept that is already implicitly present in the literature. Many problems, such as state preparation and syndrome extraction, can be naturally expressed as finding an implementable circuit that is fault-equivalent to an idealized specification. To utilize fault equivalence in a computationally tractable manner, we adapt the ZX calculus, a diagrammatic language for quantum computing. We restrict its rewrite system to not only preserve the underlying linear map but also fault equivalence, i.e. the circuit's behaviour under noise. Enabled by our framework, we verify, optimise, and synthesise new and efficient circuits for syndrome extraction and cat state preparation. We confirm the improved performance of our optimised circuits in simulation. We anticipate that fault equivalence can capture and unify different approaches in fault-tolerant quantum computing, paving the way for an end-to-end circuit compilation framework.