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Wire Codes

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AbstractQuantum information is fragile and must be protected by a quantum error-correcting code for large-scale practical applications. Recently, highly efficient quantum codes have been discovered which require a high degree of spatial connectivity. This raises the question of how to realize these codes with minimal overhead under physical hardware connectivity constraints. Here, we introduce a general recipe to transform any quantum stabilizer code into a subsystem code that has local interactions, with weight and degree three, on a given graph. We call the subsystem codes produced by our recipe wire codes, and their code parameters depend on the input code and the given graph. Wire codes can be adapted to have a local implementation on any graph that supports a low-density embedding of the input Tanner graph, with an overhead that depends on the embedding. In particular, applying our results to a stabilizer code and a subdivision of its own Tanner graph, yields a quantum weight reduction procedure with a multiplicative qubit overhead and distance reduction that are linear in the input check degree and weight, respectively. Applying our results to hypercubic lattices leads to a construction of local subsystem codes with optimal scaling code parameters in any fixed spatial dimension. Similarly, applying our results to families of expanding graphs leads to local codes on these graphs with code parameters that depend on the degree of expansion. Our results constitute a general method to construct low-overhead subsystem codes on general graphs, which can be applied to adapt highly efficient quantum error correction procedures to hardware with restricted connectivity.Featured image: A wire code based on the quantum [[5,1,3]] code.Popular summaryQuantum low-density parity-check codes present a promising and efficient path to utility-scale quantum computation. However, their implementation demands a high degree of connectivity which is a challenge to realize in leading hardware platforms. In this work, we introduce wire codes which allow arbitrary quantum codes to be implemented using the connectivity constraints of a given hardware platform at a modest overhead cost. Our mapping produces local quantum subsystem codes with optimal code parameter scaling on hypercubic lattices. Wire codes are a promising tool for adapting highly connected quantum code constructions to implementations on a wide range of hardware.► BibTeX data@article{Baspin2026wirecodes, doi = {10.22331/q-2026-04-24-2083}, url = {https://doi.org/10.22331/q-2026-04-24-2083}, title = {Wire {C}odes}, author = {Baspin, Nou{\'{e}}dyn and Williamson, Dominic}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2083}, month = apr, year = {2026} }► References [1] Peter W. Shor. ``Scheme for reducing decoherence in quantum computer memory''. Physical Review A 52, R2493 (1995). https://doi.org/10.1103/PhysRevA.52.R2493 [2] A. M. Steane. ``Error correcting codes in quantum theory''. Phys. Rev. Lett. 77, 793–797 (1996). https://doi.org/10.1103/PhysRevLett.77.793 [3] P.W. Shor. ``Fault-tolerant quantum computation''. In Proceedings of 37th Conference on Foundations of Computer Science. Pages 56–65. (1996). https://doi.org/10.1109/SFCS.1996.548464 [4] Daniel Gottesman. ``Stabilizer codes and quantum error correction'' (1997). arXiv:quant-ph/9705052. arXiv:quant-ph/9705052 [5] D. Aharonov and M. Ben-Or. ``Fault-tolerant quantum computation with constant error''. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing. Page 176–188. STOC '97New York, NY, USA (1997). Association for Computing Machinery. https://doi.org/10.1145/258533.258579 [6] Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek. ``Resilient quantum computation''. Science 279, 342–345 (1998). https://doi.org/10.1126/science.279.5349.342 [7] A Yu Kitaev. ``Quantum computations: algorithms and error correction''.

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Magdalena de la Fuente. ``Dynamical weight reduction of Pauli measurements'' (2024). arXiv:2410.12527. arXiv:2410.12527 [63] Benjamin Rodatz, Boldizsár Poór, and Aleks Kissinger. ``Floquetifying stabiliser codes with distance-preserving rewrites'' (2024). arXiv:2410.17240. arXiv:2410.17240Cited by[1] Noah Berthusen, Shi Jie Samuel Tan, Eric Huang, and Daniel Gottesman, "Adaptive Syndrome Extraction", PRX Quantum 6 3, 030307 (2025). [2] M. Sohaib Alam, Jun Zen, and Thomas R. Scruby, "Bacon-Shor Board Games", arXiv:2504.02749, (2025). [3] Junyu Fan, Matthew Steinberg, Alexander Jahn, Chunjun Cao, and Sebastian Feld, "Biased-Noise Thresholds of Zero-Rate Holographic Codes with Tensor-Network Decoding", arXiv:2408.06232, (2024). [4] Samuel Dai, Ray Li, and Eugene Tang, "Optimal Locality and Parameter Tradeoffs for Subsystem Codes", arXiv:2503.22651, (2025). [5] Lily Wang, Andy Zeyi Liu, Ray Li, Aleksander Kubica, and Shouzhen Gu, "Check-weight-constrained quantum codes: Bounds and examples", arXiv:2601.15446, (2026). [6] Andrew C. Yuan, Nouédyn Baspin, and Dominic J. Williamson, "Quantum Weight Reduction with Layer Codes", arXiv:2603.04883, (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-04-24 08:45:36). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-04-24 08:45:35: Could not fetch cited-by data for 10.22331/q-2026-04-24-2083 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractQuantum information is fragile and must be protected by a quantum error-correcting code for large-scale practical applications. Recently, highly efficient quantum codes have been discovered which require a high degree of spatial connectivity. This raises the question of how to realize these codes with minimal overhead under physical hardware connectivity constraints. Here, we introduce a general recipe to transform any quantum stabilizer code into a subsystem code that has local interactions, with weight and degree three, on a given graph. We call the subsystem codes produced by our recipe wire codes, and their code parameters depend on the input code and the given graph. Wire codes can be adapted to have a local implementation on any graph that supports a low-density embedding of the input Tanner graph, with an overhead that depends on the embedding. In particular, applying our results to a stabilizer code and a subdivision of its own Tanner graph, yields a quantum weight reduction procedure with a multiplicative qubit overhead and distance reduction that are linear in the input check degree and weight, respectively. Applying our results to hypercubic lattices leads to a construction of local subsystem codes with optimal scaling code parameters in any fixed spatial dimension. Similarly, applying our results to families of expanding graphs leads to local codes on these graphs with code parameters that depend on the degree of expansion. Our results constitute a general method to construct low-overhead subsystem codes on general graphs, which can be applied to adapt highly efficient quantum error correction procedures to hardware with restricted connectivity.Featured image: A wire code based on the quantum [[5,1,3]] code.Popular summaryQuantum low-density parity-check codes present a promising and efficient path to utility-scale quantum computation. However, their implementation demands a high degree of connectivity which is a challenge to realize in leading hardware platforms. In this work, we introduce wire codes which allow arbitrary quantum codes to be implemented using the connectivity constraints of a given hardware platform at a modest overhead cost. Our mapping produces local quantum subsystem codes with optimal code parameter scaling on hypercubic lattices. Wire codes are a promising tool for adapting highly connected quantum code constructions to implementations on a wide range of hardware.► BibTeX data@article{Baspin2026wirecodes, doi = {10.22331/q-2026-04-24-2083}, url = {https://doi.org/10.22331/q-2026-04-24-2083}, title = {Wire {C}odes}, author = {Baspin, Nou{\'{e}}dyn and Williamson, Dominic}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2083}, month = apr, year = {2026} }► References [1] Peter W. Shor. ``Scheme for reducing decoherence in quantum computer memory''. Physical Review A 52, R2493 (1995). https://doi.org/10.1103/PhysRevA.52.R2493 [2] A. M. Steane. ``Error correcting codes in quantum theory''. Phys. Rev. Lett. 77, 793–797 (1996). https://doi.org/10.1103/PhysRevLett.77.793 [3] P.W. Shor. ``Fault-tolerant quantum computation''. In Proceedings of 37th Conference on Foundations of Computer Science. Pages 56–65. (1996). https://doi.org/10.1109/SFCS.1996.548464 [4] Daniel Gottesman. ``Stabilizer codes and quantum error correction'' (1997). arXiv:quant-ph/9705052. arXiv:quant-ph/9705052 [5] D. Aharonov and M. Ben-Or. ``Fault-tolerant quantum computation with constant error''. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing. Page 176–188. STOC '97New York, NY, USA (1997). Association for Computing Machinery. https://doi.org/10.1145/258533.258579 [6] Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek. ``Resilient quantum computation''. Science 279, 342–345 (1998). https://doi.org/10.1126/science.279.5349.342 [7] A Yu Kitaev. ``Quantum computations: algorithms and error correction''.

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