Classical product code constructions for quantum Calderbank-Shor-Steane codes

Dimiter Ostrev; Davide Orsucci; Francisco Lázaro; Balazs Matuz

doi:10.22331/q-2024-07-22-1420

Classical product code constructions for quantum Calderbank-Shor-Steane codes

Dimiter Ostrev, Davide Orsucci, Francisco Lázaro, and Balazs Matuz

Institute of Communications and Navigation, German Aerospace Center (DLR), 82234 Weßling, Germany

Published:	2024-07-22, volume 8, page 1420
Eprint:	arXiv:2209.13474v2
Doi:	https://doi.org/10.22331/q-2024-07-22-1420
Citation:	Quantum 8, 1420 (2024).

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Abstract

Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both $X$ parity checks and $Z$ parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC $3$-fold product CSS code having parameters $[[512,174,8]]$ are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.

Featured image: Monte Carlo simulation of the logical error rates of 5 different CSS codes. Left: maximum likelihood decoding under erasure channel ($\mathcal{E}_\beta$). Right: Belief Propagation decoding under depolarizing channel ($\mathcal{D}_\epsilon$). See main text for details.

Popular summary

Quantum LDPC codes are promising candidates for protecting quantum states from noise during communication, storage, or computation. While the asymptotic properties of quantum LDPC codes have received considerable attention in the literature, constructions of sparse codes with block size a few tens or a few hundreds of qubits are relatively less well-explored.

In this work, we propose a new general pattern for constructing quantum CSS codes, such that both parity check matrices of the quantum code are products of classical codes, each of which is itself a tensor product of smaller components. Within this general pattern, we identify a particular [[512,174,8]] quantum CSS code that is a promising candidate for quantum error correction in the near term. This code combines many desirable properties: high rate, high ability to correct errors, low weight of the syndrome measurements, parallelizable and hardware friendly syndrome measurement circuit, and some redundancy in the syndrome measurements that allows the correction of syndrome read-out errors. Moreover, this code achieves good performance using a message passing decoder without any post-processing. As far as the authors know, prior work on quantum error correction offers examples that have some of the above desirable properties, but no prior example has all of them at the same time.

We compared the performance of our [[512,174,8]] CSS code against examples from other quantum LDPC families. For the comparison, we searched for codes with similar block size, rate, row and column weight. We found a bicycle code, a hypergraph product code, and a quantum Tanner code that match these criteria. The results of the comparison are shown in the featured image. It can be seen from the figure that our example performs better than the other three quantum LDPC codes.

Finally, we believe that the codes from our product code families may be good candidates to be experimentally implemented, in the near future, in quantum computers based on arrays of Rydberg atoms. In fact, there exists a good synergy with this quantum computing platform, as it can realize the transport of an array of atoms and, therefore, can support a fast fully-parallelized implementation of the syndrome measurements defining our CSS product codes.

► BibTeX data

@article{Ostrev2024classicalproduct,
  doi = {10.22331/q-2024-07-22-1420},
  url = {https://doi.org/10.22331/q-2024-07-22-1420},
  title = {Classical product code constructions for quantum {C}alderbank-{S}hor-{S}teane codes},
  author = {Ostrev, Dimiter and Orsucci, Davide and L{\'{a}}zaro, Francisco and Matuz, Balazs},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1420},
  month = jul,
  year = {2024}
}

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Cited by

[1] Olai Å. Mostad, Eirik Rosnes, and Hsuan-Yin Lin, "Asymptotically Good Generalized Quantum Tanner Codes", IEEE Journal on Selected Areas in Information Theory 6, 367 (2025).

[2] Olai Å. Mostad, Eirik Rosnes, and Hsuan-Yin Lin, 2025 13th International Symposium on Topics in Coding (ISTC) 1 (2025) ISBN:979-8-3315-8983-7.

[3] Olai Å. Mostad, Hsuan-Yin Lin, Eirik Rosnes, De-Shih Lee, and Ching-Yi Lai, 2025 13th International Symposium on Topics in Coding (ISTC) 1 (2025) ISBN:979-8-3315-8983-7.

[4] Jan Krzyszkowski and Marcin Niemiec, "Analysis of Surface Code Algorithms on Quantum Hardware Using the Qrisp Framework", Electronics 14 23, 4707 (2025).

[5] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

[6] Diego Forlivesi, Lorenzo Valentini, and Marco Chiani, "Logical Error Rates of XZZX and Rotated Quantum Surface Codes", IEEE Journal on Selected Areas in Communications 42 7, 1808 (2024).

[7] Diego Forlivesi, Lorenzo Valentini, and Marco Chiani, "Logical Error Rates of XZZX and Rotated Quantum Surface Codes", arXiv:2312.17057, (2023).

[8] Meng-Yuan Li, "Quantum bootstrap product codes", arXiv:2601.22363, (2026).

[9] Dimiter Ostrev, "Quantum LDPC Codes From Intersecting Subsets", IEEE Transactions on Information Theory 70 8, 5692 (2024).

[10] Aldo Cumitini, Stefano Tinelli, Balázs Matuz, Francisco Lazaro, and Luca Barletta, "Optimal Single-Shot Decoding of Quantum Codes", IEEE Communications Letters 28 6, 1243 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-07-15 12:36:18) and SAO/NASA ADS (last updated successfully 2026-07-15 12:36:19). The list may be incomplete as not all publishers provide suitable and complete citation data.

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