quantum-computing

Simple logical quantum computation with concatenated symplectic double codes

Noah Berthusen and Elijah Durso-Sabina

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AbstractThere have been significant recent advances in constructing theoretical and practical quantum error correcting codes that function well as quantum memories; however, performing fault-tolerant logical gates on these codes is less studied, and the protocols that do exist often require significant complexity. Building off the symplectic double construction, we investigate concatenated symplectic double codes, which have a rich set of logical gates implementable using only physical single-qubit gates and qubit relabeling. Combined with injected fold-transversal gates, the full Clifford group on a single codeblock is achieved through a functionally simple circuit. We perform circuit-level simulations of state preparation and quantum error correction on these codes and show that they have promising performance at near state-of-the-art physical error rates. As such, we argue that concatenated symplectic double codes are strong contenders as the underlying computational code on medium- to large-scale quantum computers.Popular summaryQuantum error correction faces an awkward tradeoff: a code can be excellent at preserving quantum information yet cumbersome to compute with. Once information is spread across many physical qubits, even simple logical operations can require elaborate fault-tolerant circuits. Here we explore a family of codes that uses symmetry to reduce this complexity. Many logical operations can be carried out using only simple single-qubit gates and permutations of the physical qubits–operations that may be especially natural in platforms where qubits can be moved or dynamically reconnected. This makes a large class of protected computations unusually inexpensive: with occasional additional resource states, Clifford operations within a code block can be performed with little more complexity than the error correction already needed to keep the information safe. Our simulations indicate that small examples can retain competitive error-correction performance despite this computational simplicity. More broadly, the work suggests a different route to fault tolerance in which instead of building complicated circuitry for every logical gate, one can design the code so that useful computation is built into its symmetries.► BibTeX data@article{Berthusen2026simplelogical, doi = {10.22331/q-2026-08-19-2195}, url = {https://doi.org/10.22331/q-2026-08-19-2195}, title = {Simple logical quantum computation with concatenated symplectic double codes}, author = {Berthusen, Noah and Durso-Sabina, Elijah}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2195}, month = aug, year = {2026} }► References [1] Daniel Gottesman. Stabilizer codes and quantum error correction. arXiv preprint arXiv:quant-ph/9705052, 1997. URL https://doi.org/10.48550/arXiv.quant-ph/9705052. https://doi.org/10.48550/arXiv.quant-ph/9705052 arXiv:quant-ph/9705052 [2] S. B. Bravyi and A. Yu. Kitaev. 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This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-08-19 08:07:34: Cannot retrieve data from ADS due to rate limitations.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. AbstractThere have been significant recent advances in constructing theoretical and practical quantum error correcting codes that function well as quantum memories; however, performing fault-tolerant logical gates on these codes is less studied, and the protocols that do exist often require significant complexity. Building off the symplectic double construction, we investigate concatenated symplectic double codes, which have a rich set of logical gates implementable using only physical single-qubit gates and qubit relabeling. Combined with injected fold-transversal gates, the full Clifford group on a single codeblock is achieved through a functionally simple circuit. We perform circuit-level simulations of state preparation and quantum error correction on these codes and show that they have promising performance at near state-of-the-art physical error rates. As such, we argue that concatenated symplectic double codes are strong contenders as the underlying computational code on medium- to large-scale quantum computers.Popular summaryQuantum error correction faces an awkward tradeoff: a code can be excellent at preserving quantum information yet cumbersome to compute with. Once information is spread across many physical qubits, even simple logical operations can require elaborate fault-tolerant circuits. Here we explore a family of codes that uses symmetry to reduce this complexity. Many logical operations can be carried out using only simple single-qubit gates and permutations of the physical qubits–operations that may be especially natural in platforms where qubits can be moved or dynamically reconnected. This makes a large class of protected computations unusually inexpensive: with occasional additional resource states, Clifford operations within a code block can be performed with little more complexity than the error correction already needed to keep the information safe. Our simulations indicate that small examples can retain competitive error-correction performance despite this computational simplicity. More broadly, the work suggests a different route to fault tolerance in which instead of building complicated circuitry for every logical gate, one can design the code so that useful computation is built into its symmetries.► BibTeX data@article{Berthusen2026simplelogical, doi = {10.22331/q-2026-08-19-2195}, url = {https://doi.org/10.22331/q-2026-08-19-2195}, title = {Simple logical quantum computation with concatenated symplectic double codes}, author = {Berthusen, Noah and Durso-Sabina, Elijah}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2195}, month = aug, year = {2026} }► References [1] Daniel Gottesman. Stabilizer codes and quantum error correction. arXiv preprint arXiv:quant-ph/9705052, 1997. URL https://doi.org/10.48550/arXiv.quant-ph/9705052. https://doi.org/10.48550/arXiv.quant-ph/9705052 arXiv:quant-ph/9705052 [2] S. B. Bravyi and A. Yu. Kitaev. 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