quantum-computing

Dynamical codes for hardware with noisy readouts

Peter-Jan H.S. Derks, Alex Townsend-Teague, Jens Eisert, Markus S. Kesselring, Oscar Higgott, and Benjamin J. Brown

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AbstractDynamical stabilizer codes may offer a practical route to large-scale quantum computation. Such codes are defined by a schedule of error-detecting measurements, which allows for flexibility in their construction. In this work, we ask how best to optimise the measurement schedule of dynamically condensed colour codes in various limits of noise bias. We take a particular focus on the setting where measurements introduce more noise than unitary and idling operations – a noise model relevant to some hardware proposals. For measurement-biased noise models, we improve code performance by strategically repeating measurements within the schedule. For unbiased or $Z$-biased noise models, we find repeating measurements offers little improvement – somewhat contrary to our expectations – and investigate why this is. To perform this analysis, we generalise a metric called the teraquop footprint to the teraquop volume. This is the product of the number of qubits and number of rounds of measurements required such that the probability of a spacelike or timelike logical error occurring is less than $10^{-12}$. In most cases, we find differences in performance are primarily due to the number of rounds of measurements required, rather than the number of qubits – emphasising the importance of using the teraquop volume in the analysis. Additionally, our results provide another example of the importance of making use of correlated errors when decoding, in that using belief matching rather than minimum-weight perfect matching can turn a worst-performing code under a given noise model into a best-performing code.Popular summaryLarge-scale quantum computers will need to protect fragile quantum information from errors. A leading approach is quantum error correction, in which errors are detected by repeatedly measuring carefully chosen properties of the quantum system. Surprisingly, not only what is measured matters, but also when and in what order these measurements are performed. In this work, we study a family of quantum error-correcting codes known as dynamical stabilizer codes, where the measurement schedule itself becomes part of the code design. We investigate how these schedules should be optimized for different hardware platforms, with a particular focus on devices where quantum measurements are significantly noisier than other operations. We find that repeating selected measurements can substantially improve performance when measurement errors dominate, while providing little benefit when other error sources are equally important. To compare different measurement schedules, we introduce the teraquop volume, a metric that combines both the number of physical qubits and the time required to suppress logical errors below a practically relevant threshold of $10^{-12}$. Using this metric, we show that both the measurement schedule and the decoding algorithm can dramatically affect the resources required for fault-tolerant quantum computation. Our results provide practical guidance for tailoring quantum error-correcting codes to the strengths and weaknesses of future quantum computing hardware.► BibTeX data@article{Derks2026dynamicalcodes, doi = {10.22331/q-2026-07-29-2176}, url = {https://doi.org/10.22331/q-2026-07-29-2176}, title = {Dynamical codes for hardware with noisy readouts}, author = {Derks, Peter-Jan H.S. and Townsend-Teague, Alex and Eisert, Jens and Kesselring, Markus S. and Higgott, Oscar and Brown, Benjamin J.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2176}, month = jul, year = {2026} }► References [1] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. J. Math. 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Magdalena de la Fuente, Josias Old, Alex Townsend-Teague, Manuel Rispler, Jens Eisert, and Markus Müller. ``The XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime''. PRX Quantum 6, 010360 (2025). arXiv:2407.08566. https://doi.org/10.1103/PRXQuantum.6.010360 arXiv:2407.08566Cited byCould not fetch Crossref cited-by data during last attempt 2026-07-29 08:50:27: Could not fetch cited-by data for 10.22331/q-2026-07-29-2176 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-07-29 08:50:28: 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. AbstractDynamical stabilizer codes may offer a practical route to large-scale quantum computation. Such codes are defined by a schedule of error-detecting measurements, which allows for flexibility in their construction. In this work, we ask how best to optimise the measurement schedule of dynamically condensed colour codes in various limits of noise bias. We take a particular focus on the setting where measurements introduce more noise than unitary and idling operations – a noise model relevant to some hardware proposals. For measurement-biased noise models, we improve code performance by strategically repeating measurements within the schedule. For unbiased or $Z$-biased noise models, we find repeating measurements offers little improvement – somewhat contrary to our expectations – and investigate why this is. To perform this analysis, we generalise a metric called the teraquop footprint to the teraquop volume. This is the product of the number of qubits and number of rounds of measurements required such that the probability of a spacelike or timelike logical error occurring is less than $10^{-12}$. In most cases, we find differences in performance are primarily due to the number of rounds of measurements required, rather than the number of qubits – emphasising the importance of using the teraquop volume in the analysis. Additionally, our results provide another example of the importance of making use of correlated errors when decoding, in that using belief matching rather than minimum-weight perfect matching can turn a worst-performing code under a given noise model into a best-performing code.Popular summaryLarge-scale quantum computers will need to protect fragile quantum information from errors. A leading approach is quantum error correction, in which errors are detected by repeatedly measuring carefully chosen properties of the quantum system. Surprisingly, not only what is measured matters, but also when and in what order these measurements are performed. In this work, we study a family of quantum error-correcting codes known as dynamical stabilizer codes, where the measurement schedule itself becomes part of the code design. We investigate how these schedules should be optimized for different hardware platforms, with a particular focus on devices where quantum measurements are significantly noisier than other operations. We find that repeating selected measurements can substantially improve performance when measurement errors dominate, while providing little benefit when other error sources are equally important. To compare different measurement schedules, we introduce the teraquop volume, a metric that combines both the number of physical qubits and the time required to suppress logical errors below a practically relevant threshold of $10^{-12}$. Using this metric, we show that both the measurement schedule and the decoding algorithm can dramatically affect the resources required for fault-tolerant quantum computation. Our results provide practical guidance for tailoring quantum error-correcting codes to the strengths and weaknesses of future quantum computing hardware.► BibTeX data@article{Derks2026dynamicalcodes, doi = {10.22331/q-2026-07-29-2176}, url = {https://doi.org/10.22331/q-2026-07-29-2176}, title = {Dynamical codes for hardware with noisy readouts}, author = {Derks, Peter-Jan H.S. and Townsend-Teague, Alex and Eisert, Jens and Kesselring, Markus S. and Higgott, Oscar and Brown, Benjamin J.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2176}, month = jul, year = {2026} }► References [1] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. J. Math. 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