Floquetifying stabiliser codes with distance-preserving rewrites
University of Oxford, Oxford, UK
| Published: | 2026-09-03, volume 10, page 2202 |
| Editor: | Narayanan Rengaswamy |
| Eprint: | arXiv:2410.17240v4 |
| Doi: | https://doi.org/10.22331/q-2026-09-03-2202 |
| Citation: | Quantum 10, 2202 (2026). |
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Abstract
Stabiliser codes with large weight measurements can be challenging to implement fault-tolerantly. To overcome this, we propose a Floquetification procedure which, given a stabiliser code, synthesises a novel Floquet code that only uses single- and two-qubit operations. Moreover, this procedure preserves the distance and number of logicals of the original code. The new Floquet code requires additional physical qubits. This overhead is linear in the weight of the largest measurement of the original code. Our method is based on the ZX calculus, a graphical language for representing and rewriting quantum circuits. However, a problem arises with the use of ZX in the context of rewriting error-correcting codes: ZX rewrites generally do not preserve code distance. Tackling this issue, we define the notion of distance-preserving rewrite that enables the transformation of error-correcting codes without changing their distance. These distance-preserving rewrites are used to decompose arbitrary weight stabiliser measurements into quantum circuits with single- and two-qubit operations. As we only use distance-preserving rewrites, we are guaranteed that a single error in the resulting circuit creates at most a single error on the data qubits. These decompositions enable us to generalise the Floquetification procedure of Townsend-Teague et al [83] to arbitrary stabiliser codes, provably preserving the distance and number of logicals of the original code.
Featured image: Example of the Floquetification procedure based on the [[4, 2, 2]] code
Popular summary
However, they are extremely sensitive to noise: data can easily be corrupted by small environmental disturbances or hardware imperfections. Because of this, large-scale quantum computing will require quantum error correction to protect the data against noise. The most well-studied class of quantum error correction codes is stabiliser codes. Recently, a new, more dynamic class of quantum error correction codes has been discovered: Floquet codes. Floquet codes promise various advantages over stabiliser codes, including more efficient data encoding and easier computations.
In this work, we show how to create novel Floquet codes from existing stabiliser codes, translating the immense progress made on existing codes to Floquet codes. For this, we introduce a method that allows us to manipulate codes while preserving their core properties, such as how much information they encode and how well they protect that information from noise. While in this work we use this method to transform quantum error correcting codes, the techniques used have potential broader applications in the design of fault-tolerant quantum computations.
► BibTeX data
► References
[1] ``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
arXiv:2203.11137
[2] ``The ZX-calculus Is Complete for Stabilizer Quantum Mechanics'' New Journal of Physics 16, 093021 (2014).
https://doi.org/10.1088/1367-2630/16/9/093021
[3] ``A Simplified Stabilizer ZX-calculus'' Open Publishing Association (2017).
https://doi.org/10.4204/EPTCS.236.1
[4] ``The Code Distance of Floquet Codes'' (2025).
https://doi.org/10.48550/arXiv.2510.05549
[5] ``Topological Quantum Distillation'' Physical Review Letters 97, 180501 (2006).
https://doi.org/10.1103/PhysRevLett.97.180501
[6] ``Unifying Flavors of Fault Tolerance with the ZX Calculus'' Quantum 8, 1379 (2024).
https://doi.org/10.22331/q-2024-06-18-1379
[7] ``ZX-calculus and Quantum Stabilizer Theory'' thesis (2019).
https://www.cs.ox.ac.uk/people/aleks.kissinger/papers/borghans-thesis.pdf
[8] ``Quantum Codes on a Lattice with Boundary'' (1998).
https://doi.org/10.48550/arXiv.quant-ph/9811052
[9] ``Good Quantum Error-Correcting Codes Exist'' Physical Review A: Atomic, Molecular, and Optical Physics 54, 1098–1105 (1996).
https://doi.org/10.1103/PhysRevA.54.1098
[10] ``Interacting Quantum Observables'' Springer (2008).
https://doi.org/10.1007/978-3-540-70583-3_25
http://personal.strath.ac.uk/ross.duncan/papers/iqo-icalp.pdf
[11] ``Determinism in the One-Way Model'' Physical Review A 74, 052310 (2006).
https://doi.org/10.1103/PhysRevA.74.052310
[12] ``Finding Flows in the One-Way Measurement Model'' Physical Review A 77, 022328 (2008).
https://doi.org/10.1103/PhysRevA.77.022328
[13] ``The ZX Calculus Is a Language for Surface Code Lattice Surgery'' Quantum 4, 218 (2020).
https://doi.org/10.22331/q-2020-01-09-218
[14] ``Spacetime Codes of Clifford Circuits'' (2023).
https://doi.org/10.48550/arXiv.2304.05943
[15] ``Designing fault-tolerant circuits using detector error models'' Quantum 9, 1905 (2025).
https://doi.org/10.22331/q-2025-11-06-1905
[16] ``A Graphical Approach to Measurement-Based Quantum Computing'' (2013).
https://doi.org/10.48550/arXiv.1203.6242
arXiv:1203.6242
[17] ``Graph-Theoretic Simplification of Quantum Circuits with the ZX-calculus'' Quantum 4, 279 (2020).
https://doi.org/10.22331/q-2020-06-04-279
[18] ``Dynamical Weight Reduction of Pauli Measurements'' (2024).
https://doi.org/10.48550/arXiv.2410.12527
[19] ``A Pair Measurement Surface Code on Pentagons'' Quantum 7, 1156 (2023).
https://doi.org/10.22331/q-2023-10-25-1156
[20] ``The Heisenberg Representation of Quantum Computers'' Proc. XXII International Colloquium on Group Theoretical Methods in Physics, 1998 32–43 (1998).
[21] ``Stabilizer Codes and Quantum Error Correction'' thesis (1997).
https://doi.org/10.7907/rzr7-dt72
[22] ``Fault-tolerant pairwise measurement-based code on eight qubits'' Phys. Rev. A 112, 042413 (2025).
https://doi.org/10.48550/arXiv.2409.13681
[23] ``Dynamically Generated Logical Qubits'' Quantum 5, 564 (2021).
https://doi.org/10.22331/q-2021-10-19-564
[24] ``Enhanced Fault-tolerance in Photonic Quantum Computing: Floquet Code Outperforms Surface Code in Tailored Architecture'' (2024).
https://doi.org/10.48550/arXiv.2410.07065
[25] ``Graphical CSS Code Transformation Using ZX Calculus'' Open Publishing Association (2023).
https://doi.org/10.4204/EPTCS.384.1
[26] ``Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics'' Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science 569–578 (2018).
https://doi.org/10.1145/3209108.3209139
arXiv:1801.10142
[27] ``Phase-Free ZX Diagrams Are CSS Codes (...or How to Graphically Grok the Surface Code)'' (2022).
https://doi.org/10.48550/arXiv.2204.14038
[28] ``Picturing Quantum Software: An Introduction to the ZX-calculus and Quantum Compilation'' Preprint (2024).
https://github.com/zxcalc/book
[29] ``Scalable Spider Nests (...or How to Graphically Grok Transversal Non-Clifford Gates)'' Open Publishing Association (2024).
https://doi.org/10.4204/EPTCS.406.4
[30] ``Unified and Generalized Approach to Quantum Error Correction'' Physical Review Letters 94, 180501 (2005).
https://doi.org/10.1103/PhysRevLett.94.180501
[31] ``$\mathrm{XYZ}$ Ruby Code: Making a Case for a Three-Colored Graphical Calculus for Quantum Error Correction in Spacetime'' PRX Quantum 6, 010360 (2025).
https://doi.org/10.1103/PRXQuantum.6.010360
[32] ``Relaxing Hardware Requirements for Surface Code Circuits Using Time-dynamics'' Quantum 7, 1172 (2023).
https://doi.org/10.22331/q-2023-11-07-1172
[33] ``On the Constant Depth Implementation of Pauli Exponentials'' npj Quantum Information 12, 82 (2026).
https://doi.org/10.1038/s41534-026-01226-x
[34] ``A Universal Completion of the ZX-calculus'' (2017).
https://doi.org/10.48550/arXiv.1706.09877
[35] ``A Unique Normal Form for Prime-Dimensional Qudit Clifford ZX-calculus'' thesis (2022).
https://www.cs.ox.ac.uk/people/aleks.kissinger/theses/poor-thesis.pdf
[36] ``The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification'' Open Publishing Association (2023).
https://doi.org/10.4204/EPTCS.384.13
[37] ``Fault Tolerance by Construction'' (2025).
https://doi.org/10.48550/arXiv.2506.17181
[38] ``Completeness for Fault Equivalence of Clifford ZX Diagrams'' (2025).
https://doi.org/10.48550/arXiv.2510.08477
[39] ``Multiple-Particle Interference and Quantum Error Correction'' Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551–2577 (1997).
https://doi.org/10.1098/rspa.1996.0136
[40] ``Floquetifying the Colour Code'' Open Publishing Association (2023).
https://doi.org/10.4204/EPTCS.384.14
[41] ``ZX-calculus for the Working Quantum Computer Scientist'' (2020).
https://doi.org/10.48550/arXiv.2012.13966
[42] ``A Near-Minimal Axiomatisation of ZX-calculus for Pure Qubit Quantum Mechanics'' 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 1–10 (2019).
https://doi.org/10.1109/LICS.2019.8785765
arXiv:1812.09114
[43] ``Planar Floquet Codes'' (2021).
https://doi.org/10.48550/arXiv.2110.05348
[44] ``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
arXiv:2203.11137
[45] ``The ZX-calculus Is Complete for Stabilizer Quantum Mechanics'' New Journal of Physics 16, 093021 (2014).
https://doi.org/10.1088/1367-2630/16/9/093021
[46] ``A Simplified Stabilizer ZX-calculus'' Open Publishing Association (2017).
https://doi.org/10.4204/EPTCS.236.1
[47] ``The Code Distance of Floquet Codes'' (2025).
https://doi.org/10.48550/arXiv.2510.05549
[48] ``Topological Quantum Distillation'' Physical Review Letters 97, 180501 (2006).
https://doi.org/10.1103/PhysRevLett.97.180501
[49] ``Unifying Flavors of Fault Tolerance with the ZX Calculus'' Quantum 8, 1379 (2024).
https://doi.org/10.22331/q-2024-06-18-1379
[50] ``ZX-calculus and Quantum Stabilizer Theory'' thesis (2019).
https://www.cs.ox.ac.uk/people/aleks.kissinger/papers/borghans-thesis.pdf
[51] ``Quantum Codes on a Lattice with Boundary'' (1998).
https://doi.org/10.48550/arXiv.quant-ph/9811052
[52] ``Good Quantum Error-Correcting Codes Exist'' Physical Review A: Atomic, Molecular, and Optical Physics 54, 1098–1105 (1996).
https://doi.org/10.1103/PhysRevA.54.1098
[53] ``Interacting Quantum Observables'' Springer (2008).
https://doi.org/10.1007/978-3-540-70583-3_25
http://personal.strath.ac.uk/ross.duncan/papers/iqo-icalp.pdf
[54] ``Determinism in the One-Way Model'' Physical Review A 74, 052310 (2006).
https://doi.org/10.1103/PhysRevA.74.052310
[55] ``Finding Flows in the One-Way Measurement Model'' Physical Review A 77, 022328 (2008).
https://doi.org/10.1103/PhysRevA.77.022328
[56] ``The ZX Calculus Is a Language for Surface Code Lattice Surgery'' Quantum 4, 218 (2020).
https://doi.org/10.22331/q-2020-01-09-218
[57] ``Spacetime Codes of Clifford Circuits'' (2023).
https://doi.org/10.48550/arXiv.2304.05943
[58] ``Designing fault-tolerant circuits using detector error models'' Quantum 9, 1905 (2025).
https://doi.org/10.22331/q-2025-11-06-1905
[59] ``A Graphical Approach to Measurement-Based Quantum Computing'' (2013).
https://doi.org/10.48550/arXiv.1203.6242
arXiv:1203.6242
[60] ``Graph-Theoretic Simplification of Quantum Circuits with the ZX-calculus'' Quantum 4, 279 (2020).
https://doi.org/10.22331/q-2020-06-04-279
[61] ``Dynamical Weight Reduction of Pauli Measurements'' (2024).
https://doi.org/10.48550/arXiv.2410.12527
[62] ``A Pair Measurement Surface Code on Pentagons'' Quantum 7, 1156 (2023).
https://doi.org/10.22331/q-2023-10-25-1156
[63] ``The Heisenberg Representation of Quantum Computers'' Proc. XXII International Colloquium on Group Theoretical Methods in Physics, 1998 32–43 (1998).
[64] ``Stabilizer Codes and Quantum Error Correction'' thesis (1997).
https://doi.org/10.7907/rzr7-dt72
[65] ``Fault-tolerant pairwise measurement-based code on eight qubits'' Phys. Rev. A 112, 042413 (2025).
https://doi.org/10.48550/arXiv.2409.13681
[66] ``Dynamically Generated Logical Qubits'' Quantum 5, 564 (2021).
https://doi.org/10.22331/q-2021-10-19-564
[67] ``Enhanced Fault-tolerance in Photonic Quantum Computing: Floquet Code Outperforms Surface Code in Tailored Architecture'' (2024).
https://doi.org/10.48550/arXiv.2410.07065
[68] ``Graphical CSS Code Transformation Using ZX Calculus'' Open Publishing Association (2023).
https://doi.org/10.4204/EPTCS.384.1
[69] ``Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics'' Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science 569–578 (2018).
https://doi.org/10.1145/3209108.3209139
arXiv:1801.10142
[70] ``Phase-Free ZX Diagrams Are CSS Codes (...or How to Graphically Grok the Surface Code)'' (2022).
https://doi.org/10.48550/arXiv.2204.14038
[71] ``Picturing Quantum Software: An Introduction to the ZX-calculus and Quantum Compilation'' Preprint (2024).
https://github.com/zxcalc/book
[72] ``Scalable Spider Nests (...or How to Graphically Grok Transversal Non-Clifford Gates)'' Open Publishing Association (2024).
https://doi.org/10.4204/EPTCS.406.4
[73] ``Unified and Generalized Approach to Quantum Error Correction'' Physical Review Letters 94, 180501 (2005).
https://doi.org/10.1103/PhysRevLett.94.180501
[74] ``$\mathrm{XYZ}$ Ruby Code: Making a Case for a Three-Colored Graphical Calculus for Quantum Error Correction in Spacetime'' PRX Quantum 6, 010360 (2025).
https://doi.org/10.1103/PRXQuantum.6.010360
[75] ``Relaxing Hardware Requirements for Surface Code Circuits Using Time-dynamics'' Quantum 7, 1172 (2023).
https://doi.org/10.22331/q-2023-11-07-1172
[76] ``On the Constant Depth Implementation of Pauli Exponentials'' npj Quantum Information 12, 82 (2026).
https://doi.org/10.1038/s41534-026-01226-x
[77] ``A Universal Completion of the ZX-calculus'' (2017).
https://doi.org/10.48550/arXiv.1706.09877
[78] ``A Unique Normal Form for Prime-Dimensional Qudit Clifford ZX-calculus'' thesis (2022).
https://www.cs.ox.ac.uk/people/aleks.kissinger/theses/poor-thesis.pdf
[79] ``The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification'' Open Publishing Association (2023).
https://doi.org/10.4204/EPTCS.384.13
[80] ``Fault Tolerance by Construction'' (2025).
https://doi.org/10.48550/arXiv.2506.17181
[81] ``Completeness for Fault Equivalence of Clifford ZX Diagrams'' (2025).
https://doi.org/10.48550/arXiv.2510.08477
[82] ``Multiple-Particle Interference and Quantum Error Correction'' Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551–2577 (1997).
https://doi.org/10.1098/rspa.1996.0136
[83] ``Floquetifying the Colour Code'' Open Publishing Association (2023).
https://doi.org/10.4204/EPTCS.384.14
[84] ``ZX-calculus for the Working Quantum Computer Scientist'' (2020).
https://doi.org/10.48550/arXiv.2012.13966
[85] ``A Near-Minimal Axiomatisation of ZX-calculus for Pure Qubit Quantum Mechanics'' 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 1–10 (2019).
https://doi.org/10.1109/LICS.2019.8785765
arXiv:1812.09114
[86] ``Planar Floquet Codes'' (2021).
https://doi.org/10.48550/arXiv.2110.05348
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[2] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).
[3] Andrey Boris Khesin, Sarah Meng Li, Boldizsár Poór, Benjamin Rodatz, John van de Wetering, and Richie Yeung, "SpiderCat: Optimal Fault-Tolerant Cat State Preparation", arXiv:2603.05391, (2026).
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[8] Andreas Bauer and Julio C. Magdalena de la Fuente, "Planar Fault-Tolerant Circuits for Non-Clifford Gates on the 2D Color Code", PRX Quantum 7 2, 020367 (2026).
[9] Peter-Jan H. S. Derks, Alex Townsend-Teague, Jens Eisert, Markus S. Kesselring, Oscar Higgott, and Benjamin J. Brown, "Dynamical codes for hardware with noisy readouts", Quantum 10, 2176 (2026).
[10] Benjamin Rodatz, Boldizsár Poór, and Aleks Kissinger, "Fault Tolerance by Construction", arXiv:2506.17181, (2025).
[11] Shoham Jacoby, Alex Retzker, and Fernando Pastawski, "Stairway Codes: Floquetifying Bivariate Bicycle Codes and Beyond", arXiv:2603.00228, (2026).
[12] M. Sohaib Alam and Eleanor Rieffel, "Dynamical logical qubits in the Bacon-Shor code", Physical Review A 112 2, 022436 (2025).
[13] Haytham McDowall-Rose, Razin A. Shaikh, and Lia Yeh, "From Fermions to Qubits: A ZX-Calculus Perspective", arXiv:2505.06212, (2025).
[14] Linnea Grans-Samuelsson, David Aasen, and Parsa Bonderson, "Fault-tolerant pairwise measurement-based code on eight qubits", Physical Review A 112 4, 042413 (2025).
[15] Rajath Radhakrishnan, Adar Sharon, and Nathanan Tantivasadakarn, "Unveiling dynamical quantum error correcting codes via non-invertible symmetries", arXiv:2510.09565, (2025).
[16] Mateusz Kupper, Dominic Horsman, Chris Heunen, and Niel de Beaudrap, "String Diagrams for Defect-Based Surface Code Computing", arXiv:2508.14672, (2025).
[17] Boldizsár Poór, Razin A. Shaikh, and Quanlong Wang, "ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces", arXiv:2405.10896, (2024).
[18] Kwok Ho Wan, Zhenghao Zhong, and Ainhoa Zapirain, "Pauli web of the $|Y\rangle$ state surface code injection", arXiv:2501.15566, (2025).
[19] Boldizsár Poór, Benjamin Rodatz, and Aleks Kissinger, "Ultra low overhead syndrome extraction for the Steane code", Quantum Science and Technology 11 4, 045009 (2026).
[20] Cory T. Aitchison and Benjamin Béri, "Spacetime Spins: Statistical mechanics for error correction with stabilizer circuits", arXiv:2512.21991, (2025).
[21] Jahan Claes, "Dynamic circuit for the honeycomb Floquet code", Physical Review A 112 6, 062406 (2025).
[22] Bob Coecke, Aleks Kissinger, Stefano Gogioso, Selma Dündar-Coecke, Caterina Puca, Lia Yeh, Muhammad Hamza Waseem, Emmanuel M. Pothos, Sieglinde Pfaendler, Vincent Wang-Mascianica, Thomas Cervoni, Ferdi Tomassini, Vincent Anandraj, Peter Sigrist, and Ilyas Khan, "High schoolers excel at Oxford quantum course using pictorial mathematics", arXiv:2512.00141, (2025).
[23] Yichen Xu and Arpit Dua, "Fault-tolerant protocols through spacetime concatenation", arXiv:2504.08918, (2025).
[24] Peter-Jan H. S. Derks, Alex Townsend-Teague, Ansgar G. Burchards, and Jens Eisert, "Designing fault-tolerant circuits using detector error models", Quantum 9, 1905 (2025).
[25] Kwok Ho Wan, "Iteratively decoded magic state distillation", arXiv:2410.17992, (2024).
[26] Jelena Mackeprang and Jonas Helsen, "A Bravyi-König theorem for Floquet codes generated by locally conjugate instantaneous stabiliser groups", arXiv:2601.21863, (2026).
[27] Kwok Ho Wan and Zhenghao Zhong, "Pauli webs spun by transversal $|Y\rangle$ state initialisation", arXiv:2502.00957, (2025).
[28] Tom Peham, Erik Weilandt, and Robert Wille, "Optimizing fault-tolerant cat state preparation", Physical Review A 113 6, 062426 (2026).
[29] Giovanni de Felice, Boldizsár Poór, Cole Comfort, Lia Yeh, Mateusz Kupper, William Cashman, and Bob Coecke, "A dataflow programming framework for linear optical distributed quantum computing", arXiv:2601.08389, (2026).
[30] Maximilian Schweikart, Linnea Grans-Samuelsson, Aleks Kissinger, and Benjamin Rodatz, "Preserving MWPM-Decodability in Fault-Equivalent Rewrites", arXiv:2603.19522, (2026).
[31] Bob Coecke, Aleks Kissinger, Stefano Gogioso, Selma Dündar-Coecke, Caterina Puca, Lia Yeh, Muhammad Hamza Waseem, Emmanuel M. Pothos, Sieglinde Pfaendler, Vincent Wang-Mascianica, Thomas Cervoni, Ferdi Tomassini, Vincent Anandraj, Peter Sigrist, and Ilyas Khan, "High schoolers excel at Oxford quantum course using pictorial mathematics", Journal of Physics Conference Series 3189 1, 012008 (2026).
[32] Giovanni de Felice, Boldizsár Poór, Cole Comfort, Lia Yeh, Mateusz Kupper, William Cashman, and Bob Coecke, "A dataflow programming framework for linear optical distributed quantum computing", Quantum 10, 1972 (2026).
The above citations are from SAO/NASA ADS (last updated successfully 2026-09-07 20:17:22). The list may be incomplete as not all publishers provide suitable and complete citation data.
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