Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities

Diego Forlivesi; Lorenzo Valentini; Marco Chiani

doi:10.22331/q-2025-12-16-1950

Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities

Diego Forlivesi, Lorenzo Valentini, and Marco Chiani

Department of Electrical, Electronic, and Information Engineering ``Guglielmo Marconi'' and CNIT/WiLab, University of Bologna, 40136 Bologna, Italy.

Published:	2025-12-16, volume 9, page 1950
Editor:	Philipp Schindler
Eprint:	arXiv:2305.01301v4
Doi:	https://doi.org/10.22331/q-2025-12-16-1950
Citation:	Quantum 9, 1950 (2025).

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Abstract

We analyze the performance of quantum stabilizer codes, one of the most important classes for practical implementations, on both symmetric and asymmetric quantum channels. To this aim, we first derive the weight enumerator (WE) for the undetectable errors based on the quantum MacWilliams identities. The WE is then used to evaluate tight upper bounds on the error rate of CSS quantum codes with minimum weight decoding. For surface codes we also derive a simple closed form expression of the bounds over the depolarizing channel. We introduce a novel approach that combines the knowledge of WE with a logical operator analysis, allowing the derivation of the exact asymptotic error rate for short codes. For example, on a depolarizing channel with physical error rate $\rho \to 0$, the logical error rate $\rho_\mathrm{L}$ is asymptotically $\rho_\mathrm{L} \approx 16 \rho^2$ for the $[[9,1,3]]$ Shor code, $\rho_\mathrm{L} \approx 16.3 \rho^2$ for the $[[7,1,3]]$ Steane code, $\rho_\mathrm{L} \approx 18.7 \rho^2$ for the $[[13,1,3]]$ surface code, and $\rho_\mathrm{L} \approx 149.3 \rho^3$ for the $[[41,1,5]]$ surface code. For larger codes our bound provides $\rho_\mathrm{L} \approx 1215 \rho^4$ and $\rho_\mathrm{L} \approx 663 \rho^5$ for the $[[85,1,7]]$ and the $[[181,1,10]]$ surface codes, respectively. Finally, we extend our analysis to include realistic, noisy syndrome extraction circuits by modeling error propagation throughout gadgets. This enables estimation of logical error rates under faulty measurements. The performance analysis serves as a design tool for developing fault-tolerant quantum systems by guiding the selection of quantum codes based on their error correction capability. Additionally, it offers a novel perspective on quantum degeneracy, showing it represents the fraction of non-correctable error patterns shared by multiple logical operators.

Featured image: Example of errors leading to a logical operator of weight $w=4$ for the $[[ 13,1,3 ]]$ surface code.

Popular summary

The exploitation of the unique features of quantum mechanics has opened new perspectives on how we can sense, process, and communicate information. From an engineering point of view, there are many challenges to solve, calling for both theoretical and experimental research studies. The aim is to progress towards the already known possible applications of quantum information technologies, as well as those currently still unforeseen, that will arise when practical implementations become available. One of the main challenges is how to deal with the noise caused by unwanted interaction of the quantum information with the environment. Quantum error correcting codes, where a redundant representation of quantum states protects from certain types of errors, are therefore of paramount importance for quantum computation, quantum memories, and quantum communication systems. In this paper we provide an analytical evaluation of the performance of stabilizer codes, like quantum low-density parity-check (QLDPC) codes and surface codes. We propose a framework for the performance investigation of stabilizer codes by means of the quantum MacWilliams identities. Moreover, we develop a logical operator analysis leading to exact expressions for the logical error rates, assuming complete decoders (decoders that always attempt to correct the error). Specifically, we analyze minimum weight (MW) decoding, which finds the lowest weight error consistent with the syndrome. The analysis is conducted for both symmetric and asymmetric models of quantum channel errors. In practical quantum systems, however, syndrome extraction is a critical yet error-prone component of quantum error correction. Measurements are inherently noisy and typically require repetition to ensure reliability. Additionally, faults during extraction can propagate, causing high weight correlated errors. Thus, we introduce a framework that models the full syndrome extraction process, incorporating gate-specific noise and measurement imperfections. This enables the estimation of logical error rates under realistic circuit-level noise assumptions.
The key contributions of the paper can be summarized as follows:
– we derive the weight enumerator L(z) for the undetectable errors of arbitrary stabilizer codes via MacWilliams identities;
– we derive theoretical upper bounds for the error correction capability of CSS stabilizer codes;
– we derive closed form expressions for the coefficients of L(z) which significantly impact the performance of surface codes for any code distance;
– we derive the exact performance of stabilizer codes under MW decoding, including surface codes under MWPM decoding, over symmetric and asymmetric channels;
– we introduce a novel perspective on quantum degeneracy, analyzing its influence on the error correction capability of a quantum code.

► BibTeX data

@article{Forlivesi2025performanceanalysis,
  doi = {10.22331/q-2025-12-16-1950},
  url = {https://doi.org/10.22331/q-2025-12-16-1950},
  title = {Performance {A}nalysis of {Q}uantum {CSS} {E}rror-{C}orrecting {C}odes via {M}ac{W}illiams {I}dentities},
  author = {Forlivesi, Diego and Valentini, Lorenzo and Chiani, Marco},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1950},
  month = dec,
  year = {2025}
}

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