quantum-computing

Error Correction in Dynamical Codes

Esther Xiaozhen Fu and Daniel Gottesman

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AbstractWe ask what is the general framework for a quantum error correcting code that is defined by a sequence of measurements. Recently, there has been much interest in Floquet codes and space-time codes. In this work, we define and study the distance of a dynamical code. This is a subtle concept and difficult to determine: At any given time, the system will be in a subspace which forms a quantum error-correcting code with a given distance, but the full error correction capability of that code may not be available due to the schedule of measurements associated with the code. We address this challenge by developing an algorithm that tracks information we have learned about the error syndromes through the protocol and put that together to determine the distance of a dynamical code, in a non-fault-tolerant context. We use the tools developed for the algorithm to analyze the initialization and masking properties of a generic Floquet code. Further, we look at properties of dynamical codes under the constraint of geometric locality with a view to understand whether the fundamental limitations on logical gates and code parameters imposed by geometric locality for traditional codes can be surpassed in the dynamical paradigm. We find that codes with a limited number of long range connectivity will not allow non-Clifford gates to be implemented with finite depth circuits in the 2D setting. Popular summaryConsider the class of quantum error correcting codes that are defined by a sequence of measurements. What is the error correcting capability of a code that depends on the scheduling of the measurements associated with it? What are the fundamental limitations on logical gates and code parameters for this class? The existing theoretical framework and code properties are only studied in the context of traditional stabilizer codes with fixed stabilizer generators. In this paper, we take a step towards laying down the theoretical framework for dynamical codes. We introduce a classification of stabilizer generators by how accessible their syndrome information is given a measurement sequence. The distance of a dynamical code is then formalized based on its syndrome accessibility. We also present a classically efficient algorithm that fully classifies the stabilizers and obtains syndrome information for a dynamical code. Our results are applied to study initialization of a generic Floquet codes, revealing the underlying structure of an arbitrary Floquet code.► BibTeX data@article{Fu2025errorcorrectionin, doi = {10.22331/q-2025-10-20-1886}, url = {https://doi.org/10.22331/q-2025-10-20-1886}, title = {Error {C}orrection in {D}ynamical {C}odes}, author = {Fu, Esther Xiaozhen and Gottesman, Daniel}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1886}, month = oct, year = {2025} }► References [1] Matthew B. Hastings and Jeongwan Haah. ``Dynamically Generated Logical Qubits''. Quantum 5, 564 (2021). https://doi.org/10.22331/q-2021-10-19-564 [2] Jeongwan Haah and Matthew B. Hastings. ``Boundaries for the Honeycomb Code''. Quantum 6, 693 (2022). https://doi.org/10.22331/q-2022-04-21-693 [3] Christophe Vuillot. ``Planar floquet codes'' (2021). arXiv:2110.05348. https://doi.org/10.48550/arXiv.2110.05348 arXiv:2110.05348 [4] Markus S. Kesselring, Julio C. Magdalena de la Fuente, Felix Thomsen, Jens Eisert, Stephen D. Bartlett, and Benjamin J. Brown. ``Anyon condensation and the color code'' (2024). PRX Quantum 5, 010342. https://doi.org/10.1103/PRXQuantum.5.010342 [5] Tyler D. Ellison, Joseph Sullivan, and Arpit Dua. ``Floquet codes with a twist'' (2023). arXiv:2306.08027. https://doi.org/10.48550/arXiv.2306.08027 arXiv:2306.08027 [6] Ali Fahimniya, Hossein Dehghani, Kishor Bharti, Sheryl Mathew, Alicia J. Kollár, Alexey V. Gorshkov, and Michael J. Gullans. ``Fault-tolerant hyperbolic floquet quantum error correcting codes'' (2025). Quantum 9, 1849. https://doi.org/10.22331/q-2025-09-05-1849 [7] Oscar Higgott and Nikolas P. Breuckmann. ``Constructions and performance of hyperbolic and semi-hyperbolic floquet codes'' (2024). PRX Quantum 5, 040327. https://doi.org/10.1103/PRXQuantum.5.040327 [8] Zhehao Zhang, David Aasen, and Sagar Vijay. ``$x$-cube floquet code: A dynamical quantum error correcting code with a subextensive number of logical qubits''. Phys. Rev. B 108, 205116 (2023). https://doi.org/10.1103/PhysRevB.108.205116 [9] Andreas Bauer. ``Topological error correcting processes from fixed-point path integrals'' (2024). Quantum 8, 1288. https://doi.org/10.22331/q-2024-03-20-1288 [10] Margarita Davydova, Nathanan Tantivasadakarn, and Shankar Balasubramanian. ``Floquet codes without parent subsystem codes''. PRX Quantum 4, 020341 (2023). https://doi.org/10.1103/PRXQuantum.4.020341 [11] Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton. ``A fault-tolerant honeycomb memory''. Quantum 5, 605 (2021). https://doi.org/10.22331/q-2021-12-20-605 [12] Adam Paetznick, Christina Knapp, Nicolas Delfosse, Bela Bauer, Jeongwan Haah, Matthew B. Hastings, and Marcus P. da Silva. ``Performance of planar floquet codes with majorana-based qubits''. PRX Quantum 4, 010310 (2023). https://doi.org/10.1103/PRXQuantum.4.010310 [13] David Aasen, Jeongwan Haah, Zhi Li, and Roger S. K. Mong. ``Measurement quantum cellular automata and anomalies in floquet codes'' (2023). arXiv:2304.01277. https://doi.org/10.48550/arXiv.2304.01277 arXiv:2304.01277 [14] David Aasen, Zhenghan Wang, and Matthew B. Hastings. ``Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes''. Physical Review B 106, 085122 (2022). https://doi.org/10.1103/PhysRevB.106.085122 [15] Joseph Sullivan, Rui Wen, and Andrew C. Potter. ``Floquet codes and phases in twist-defect networks'' (2023). Phys. Rev. B 108, 195134. https://doi.org/10.1103/PhysRevB.108.195134 [16] M. Sohaib Alam and Eleanor Rieffel. ``Dynamical logical qubits in the bacon-shor code'' (2025). Phys. Rev. A 112, 022436. https://doi.org/10.1103/nfxv-3dp7 [17] Yichen Xu and Arpit Dua. ``Fault-tolerant protocols through spacetime concatenation'' (2025). arXiv:2504.08918. https://doi.org/10.48550/arXiv.2504.08918 arXiv:2504.08918 [18] Cupjin Huang and Michael Newman. ``Transversal switching between generic stabilizer codes'' (2018). arXiv:1709.09282. https://doi.org/10.48550/arXiv.1709.09282 arXiv:1709.09282 [19] Aleksander Kubica and Michael E. Beverland. ``Universal transversal gates with color codes - a simplified approach''. Physical Review A 91, 032330 (2015). https://doi.org/10.1103/PhysRevA.91.032330 [20] Jonas T. Anderson, Guillaume Duclos-Cianci, and David Poulin. ``Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes''.

Physical Review Letters 113, 080501 (2014). https://doi.org/10.1103/PhysRevLett.113.080501 [21] Bryan Eastin and Emanuel Knill. ``Restrictions on Transversal Encoded Quantum Gate Sets''.

Physical Review Letters 102, 110502 (2009). https://doi.org/10.1103/PhysRevLett.102.110502 [22] Sergey Bravyi. ``Subsystem codes with spatially local generators''. Physical Review A 83, 012320 (2011). https://doi.org/10.1103/PhysRevA.83.012320 [23] Daniel Gottesman. ``Opportunities and challenges in fault-tolerant quantum computation'' (2022). arXiv:2210.15844. https://doi.org/10.48550/arXiv.2210.15844 arXiv:2210.15844 [24] Nicolas Delfosse and Adam Paetznick. ``Spacetime codes of clifford circuits'' (2023). arXiv:2304.05943. https://doi.org/10.48550/arXiv.2304.05943 arXiv:2304.05943 [25] Dave Bacon, Steven T. Flammia, Aram W. Harrow, and Jonathan Shi. ``Sparse Quantum Codes from Quantum Circuits''. IEEE Transactions on Information Theory 63, 2464–2479 (2017). https://doi.org/10.1109/TIT.2017.2663199 [26] Peter-Jan H. S. Derks, Alex Townsend-Teague, Ansgar G. Burchards, and Jens Eisert. ``Designing fault-tolerant circuits using detector error models'' (2024). arXiv:2407.13826. https://doi.org/10.48550/arXiv.2407.13826 arXiv:2407.13826 [27] Michael E. Beverland, Shilin Huang, and Vadym Kliuchnikov. ``Fault tolerance of stabilizer channels'' (2024). arXiv:2401.12017. https://doi.org/10.48550/arXiv.2401.12017 arXiv:2401.12017 [28] Sergey Bravyi and Barbara Terhal. ``A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes''. New Journal of Physics 11, 043029 (2009). https://doi.org/10.1088/1367-2630/11/4/043029 [29] Sergey Bravyi and Robert Koenig. ``Classification of topologically protected gates for local stabilizer codes''.

Physical Review Letters 110, 170503 (2013). https://doi.org/10.1103/PhysRevLett.110.170503 [30] Fernando Pastawski and Beni Yoshida. ``Fault-tolerant logical gates in quantum error-correcting codes''. Physical Review A 91, 012305 (2015). https://doi.org/10.1103/PhysRevA.91.012305 [31] Dave Bacon. ``Operator quantum error-correcting subsystems for self-correcting quantum memories''. Physical Review A 73 (2006). https://doi.org/10.1103/physreva.73.012340 [32] Sergey Bravyi, Guillaume Duclos-Cianci, David Poulin, and Martin Suchara. ``Subsystem surface codes with three-qubit check operators'' (2013). arXiv:1207.1443. https://doi.org/10.48550/arXiv.1207.1443 arXiv:1207.1443 [33] Margarita Davydova, Nathanan Tantivasadakarn, Shankar Balasubramanian, and David Aasen. ``Quantum computation from dynamic automorphism codes'' (2024). Quantum 8, 1448. https://doi.org/10.22331/q-2024-08-27-1448 [34] Sergey Bravyi, David Poulin, and Barbara Terhal. ``Tradeoffs for Reliable Quantum Information Storage in 2D Systems''.

Physical Review Letters 104, 050503 (2010). https://doi.org/10.1103/PhysRevLett.104.050503 [35] Nouédyn Baspin, Venkatesan Guruswami, Anirudh Krishna, and Ray Li. ``Improved rate-distance trade-offs for quantum codes with restricted connectivity'' (2023). Quantum Science and Technology 9, 045024 (2024). https://doi.org/10.1088/2058-9565/ad8370 [36] Sergey Bravyi and Andrew Cross. ``Doubled color codes'' (2015). arXiv:1509.03239. https://doi.org/10.48550/arXiv.1509.03239 arXiv:1509.03239 [37] Nouédyn Baspin and Anirudh Krishna. ``Quantifying nonlocality: how outperforming local quantum codes is expensive''.

Physical Review Letters 129, 050505 (2022). https://doi.org/10.1103/PhysRevLett.129.050505Cited byOn Crossref's cited-by service no data on citing works was found (last attempt 2025-10-23 21:26:26). Could not fetch ADS cited-by data during last attempt 2025-10-23 21:26:26: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractWe ask what is the general framework for a quantum error correcting code that is defined by a sequence of measurements. Recently, there has been much interest in Floquet codes and space-time codes. In this work, we define and study the distance of a dynamical code. This is a subtle concept and difficult to determine: At any given time, the system will be in a subspace which forms a quantum error-correcting code with a given distance, but the full error correction capability of that code may not be available due to the schedule of measurements associated with the code. We address this challenge by developing an algorithm that tracks information we have learned about the error syndromes through the protocol and put that together to determine the distance of a dynamical code, in a non-fault-tolerant context. We use the tools developed for the algorithm to analyze the initialization and masking properties of a generic Floquet code. Further, we look at properties of dynamical codes under the constraint of geometric locality with a view to understand whether the fundamental limitations on logical gates and code parameters imposed by geometric locality for traditional codes can be surpassed in the dynamical paradigm. We find that codes with a limited number of long range connectivity will not allow non-Clifford gates to be implemented with finite depth circuits in the 2D setting. Popular summaryConsider the class of quantum error correcting codes that are defined by a sequence of measurements. What is the error correcting capability of a code that depends on the scheduling of the measurements associated with it? What are the fundamental limitations on logical gates and code parameters for this class? The existing theoretical framework and code properties are only studied in the context of traditional stabilizer codes with fixed stabilizer generators. In this paper, we take a step towards laying down the theoretical framework for dynamical codes. We introduce a classification of stabilizer generators by how accessible their syndrome information is given a measurement sequence. The distance of a dynamical code is then formalized based on its syndrome accessibility. We also present a classically efficient algorithm that fully classifies the stabilizers and obtains syndrome information for a dynamical code. Our results are applied to study initialization of a generic Floquet codes, revealing the underlying structure of an arbitrary Floquet code.► BibTeX data@article{Fu2025errorcorrectionin, doi = {10.22331/q-2025-10-20-1886}, url = {https://doi.org/10.22331/q-2025-10-20-1886}, title = {Error {C}orrection in {D}ynamical {C}odes}, author = {Fu, Esther Xiaozhen and Gottesman, Daniel}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1886}, month = oct, year = {2025} }► References [1] Matthew B. Hastings and Jeongwan Haah. ``Dynamically Generated Logical Qubits''. Quantum 5, 564 (2021). https://doi.org/10.22331/q-2021-10-19-564 [2] Jeongwan Haah and Matthew B. Hastings. ``Boundaries for the Honeycomb Code''. Quantum 6, 693 (2022). https://doi.org/10.22331/q-2022-04-21-693 [3] Christophe Vuillot. ``Planar floquet codes'' (2021). arXiv:2110.05348. https://doi.org/10.48550/arXiv.2110.05348 arXiv:2110.05348 [4] Markus S. Kesselring, Julio C. Magdalena de la Fuente, Felix Thomsen, Jens Eisert, Stephen D. Bartlett, and Benjamin J. Brown. ``Anyon condensation and the color code'' (2024). PRX Quantum 5, 010342. https://doi.org/10.1103/PRXQuantum.5.010342 [5] Tyler D. Ellison, Joseph Sullivan, and Arpit Dua. ``Floquet codes with a twist'' (2023). arXiv:2306.08027. https://doi.org/10.48550/arXiv.2306.08027 arXiv:2306.08027 [6] Ali Fahimniya, Hossein Dehghani, Kishor Bharti, Sheryl Mathew, Alicia J. Kollár, Alexey V. Gorshkov, and Michael J. Gullans. ``Fault-tolerant hyperbolic floquet quantum error correcting codes'' (2025). Quantum 9, 1849. https://doi.org/10.22331/q-2025-09-05-1849 [7] Oscar Higgott and Nikolas P. Breuckmann. ``Constructions and performance of hyperbolic and semi-hyperbolic floquet codes'' (2024). PRX Quantum 5, 040327. https://doi.org/10.1103/PRXQuantum.5.040327 [8] Zhehao Zhang, David Aasen, and Sagar Vijay. ``$x$-cube floquet code: A dynamical quantum error correcting code with a subextensive number of logical qubits''. Phys. Rev. B 108, 205116 (2023). https://doi.org/10.1103/PhysRevB.108.205116 [9] Andreas Bauer. ``Topological error correcting processes from fixed-point path integrals'' (2024). Quantum 8, 1288. https://doi.org/10.22331/q-2024-03-20-1288 [10] Margarita Davydova, Nathanan Tantivasadakarn, and Shankar Balasubramanian. ``Floquet codes without parent subsystem codes''. PRX Quantum 4, 020341 (2023). https://doi.org/10.1103/PRXQuantum.4.020341 [11] Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton. ``A fault-tolerant honeycomb memory''. Quantum 5, 605 (2021). https://doi.org/10.22331/q-2021-12-20-605 [12] Adam Paetznick, Christina Knapp, Nicolas Delfosse, Bela Bauer, Jeongwan Haah, Matthew B. Hastings, and Marcus P. da Silva. ``Performance of planar floquet codes with majorana-based qubits''. PRX Quantum 4, 010310 (2023). https://doi.org/10.1103/PRXQuantum.4.010310 [13] David Aasen, Jeongwan Haah, Zhi Li, and Roger S. K. Mong. ``Measurement quantum cellular automata and anomalies in floquet codes'' (2023). arXiv:2304.01277. https://doi.org/10.48550/arXiv.2304.01277 arXiv:2304.01277 [14] David Aasen, Zhenghan Wang, and Matthew B. Hastings. ``Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes''. Physical Review B 106, 085122 (2022). https://doi.org/10.1103/PhysRevB.106.085122 [15] Joseph Sullivan, Rui Wen, and Andrew C. Potter. ``Floquet codes and phases in twist-defect networks'' (2023). Phys. Rev. B 108, 195134. https://doi.org/10.1103/PhysRevB.108.195134 [16] M. Sohaib Alam and Eleanor Rieffel. ``Dynamical logical qubits in the bacon-shor code'' (2025). Phys. Rev. A 112, 022436. https://doi.org/10.1103/nfxv-3dp7 [17] Yichen Xu and Arpit Dua. ``Fault-tolerant protocols through spacetime concatenation'' (2025). arXiv:2504.08918. https://doi.org/10.48550/arXiv.2504.08918 arXiv:2504.08918 [18] Cupjin Huang and Michael Newman. ``Transversal switching between generic stabilizer codes'' (2018). arXiv:1709.09282. https://doi.org/10.48550/arXiv.1709.09282 arXiv:1709.09282 [19] Aleksander Kubica and Michael E. Beverland. ``Universal transversal gates with color codes - a simplified approach''. Physical Review A 91, 032330 (2015). https://doi.org/10.1103/PhysRevA.91.032330 [20] Jonas T. Anderson, Guillaume Duclos-Cianci, and David Poulin. ``Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes''.

Physical Review Letters 113, 080501 (2014). https://doi.org/10.1103/PhysRevLett.113.080501 [21] Bryan Eastin and Emanuel Knill. ``Restrictions on Transversal Encoded Quantum Gate Sets''.

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