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  • Open Access

Universal Graph Representation of Stabilizer Codes

Andrey Boris Khesin*,†, Jonathan Z. Lu‡, and Peter W. Shor

  • *Contact author: khesin@mit.edu
  • †Presently at Department of Computer Science, University of Oxford, Oxford OX1 3QG, United Kingdom.
  • ‡Contact author: lujz@mit.edu
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PRX Quantum 6, 040325 – Published 5 November, 2025

DOI: https://doi.org/10.1103/1gjs-2rhx

Abstract

While stabilizer tableaus have proven useful as a descriptive tool for additive quantum codes, they otherwise offer little guidance for concrete constructions or algorithm analysis. We introduce a representation of stabilizer codes as graphs with certain structures, and prove via the ZX calculus that this representation is related to stabilizer tableaus by an efficiently computable bijection. This gives a new universal recipe for code construction by way of finding graphs with nice properties. The graph representation gives insight into both code construction and algorithms. We construct as examples families of ⟦n,Θ(nlogn),Θ(logn)⟧ and ⟦n,Ω(n4/5),Θ(n1/5)⟧ codes. We use graphs in a probabilistic analysis to extend the quantum Gilbert-Varshamov bound into a three-way distance-rate-weight trade-off. Moreover, code properties such as distance and encoding circuit depth are bounded by simple functions of the graph degree. We prove that key coding algorithms—distance approximation, minimum-weight generator selection, and decoding—are unified as instances of one optimization game on a graph. By studying this game, we construct an efficient greedy decoder and prove that it corrects all recoverable errors for all graphs with cycle lengths no shorter than 13 (reducible to 5 with mild extra constraints); these include the above two families. Our results suggest that graphs are generically useful for the study of stabilizer codes.

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Physics Subject Headings (PhySH)

Popular Summary

To build quantum computers which can reliably execute algorithms, we must first design good quantum error-correcting codes. Stabilizer codes are a popular class of quantum codes, being the quantum analog to classical linear codes. Yet compared to the classical setting, we understand little about stabilizer codes in general—how to construct them, how to control their check weight, and how to decode them. This work uses graphs to begin answering such questions. We transform every stabilizer code into a graph. Then, we show how graph properties correspond to code properties and use this correspondence to design interesting codes as well as a decoding algorithm which provably succeeds if the graph satisfies simple properties.

Our approach is well motivated from a classical standpoint: much of our understanding of classical codes comes from representing them with an object known as Tanner graphs. However, Tanner graphs do not naturally generalize to represent all stabilizer codes. Instead, we constructed a completely new graph representation. We then use graphs to construct codes and design algorithms. Notably, we show that finding the code distance, selecting low-weight parity checks, and decoding are all strategies for a single unified game. We study this game to build an efficient decoder and prove that if the graph lacks short cycles, the decoder succeeds.

This work shows that graphs give much insight into stabilizer code construction and algorithms. Future directions include graph algorithms to find low-weight checks, improved constructions, and new graph representations which are codesigned for more specific code problems.

Article Text

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Outline

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PRX Quantum 6, 040325– Published 5 November, 2025
  • Received 14 May 2025
  • Revised 11 August 2025
  • Accepted 25 September 2025
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