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arxiv: 2606.02542 · v1 · pith:2C74PJKOnew · submitted 2026-06-01 · 🪐 quant-ph

Chutes and Ladders: Dynamical Automorphisms via the ZX-Calculus

Alexander Frei , Sascha Zakaib-Bernier , Zachary Mann , Michael Vasmer , Victor V. Albert This is my paper

Pith reviewed 2026-06-28 14:17 UTC · model grok-4.3

classification 🪐 quant-ph
keywords ZX-calculusstabilizer codesgauge fixingcode switchingdynamical automorphismslogical Clifford gatesseven-qubit codeFloquet codes
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The pith

Closed loops of gauge-fixing steps implement logical Clifford gates via the ZX-calculus.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the ZX-calculus to dynamical stabilizer codes by using the link between measurement-based code switching and gauge fixing. It shows that sequences of gauge-fixing operations can be chained into closed paths that return to the original codespace while applying a non-trivial logical Clifford gate. These paths act as shortcuts, called chutes and ladders, compared with sequences of single-qubit Cliffords and permutations. A reader would care because the construction supplies a graphical, machine-interpretable route to logical gates that are otherwise unavailable or difficult to locate, with an explicit example of a logical phase gate on the seven-qubit bare code.

Core claim

We combine gauge-fixing steps to implement a closed loop in the space of stabilizer codes, returning to the original codespace up to a logical Clifford gate. These measurement-based paths in the space of stabilizer codes can be viewed as shortcuts, or chutes and ladders, relative to single-qubit Clifford operations and qubit permutations. This yields a machine-interpretable method for constructing dynamical automorphisms. As an example, we implement a logical phase gate via distance-preserving code switching for the seven-qubit bare code, which has no non-trivial logical Clifford gates based on single-qubit Clifford operations and qubit permutations.

What carries the argument

Closed loops formed by combining gauge-fixing steps in the space of stabilizer codes, which realize dynamical automorphisms inside the ZX-calculus.

If this is right

  • Logical Clifford gates become constructible for stabilizer codes that admit no such gates under single-qubit operations and permutations alone.
  • The same loop-construction technique applies directly to Floquet and other dynamical codes.
  • The graphical language makes systematic search for desired logical-gate implementations feasible.
  • Dynamical automorphisms can be generated and verified by machine without manual circuit rewriting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be automated inside ZX software to enumerate gate sets for families of codes beyond the seven-qubit example.
  • If the loops preserve distance, they might serve as building blocks for fault-tolerant gate teleportation protocols.
  • Similar closed paths might be definable in other graphical calculi, connecting the construction to broader diagrammatic quantum computation.
  • The approach reframes code switching as a computational primitive rather than solely an error-correction tool.

Load-bearing premise

The link between measurement-based code switching and gauge fixing permits gauge-fixing steps to be assembled into closed loops that apply non-trivial logical Clifford gates within the ZX-calculus.

What would settle it

A concrete counter-example would be a ZX-calculus diagram for the described gauge-fixing loop on the seven-qubit code that either fails to return to the original codespace or applies only the identity instead of the claimed logical phase gate.

Figures

Figures reproduced from arXiv: 2606.02542 by Alexander Frei, Michael Vasmer, Sascha Zakaib-Bernier, Victor V. Albert, Zachary Mann.

Figure 1
Figure 1. Figure 1: FIG. 1. ZX-encoding graphs of subsystem codes that gauge fix into a stabilizer code. From left to right, we have the ZX [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dynamical automorphism of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Local Clifford (LC) orbit of the 3-qubit code from [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Gauge fixing examples and dynamical automorphism [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Dynamical automorphism for the bare code implementing a logical [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Rules of the ZX-calculus. (a) Spider fusion. (b) Color change. (c) Identity. (d) [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Example of an encoding circuit of a subsystem code. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Clifford conversion sequence from the [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. How to easily read off the Steane code from the conversion sequence ( [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Code deformations based on [ [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
read the original abstract

The ZX-calculus is a powerful graphical language for manipulating quantum circuits, which has recently found many applications in quantum error correction. We extend this language to handle Floquet and other dynamical stabilizer codes via the connection between measurement-based code switching and gauge fixing (arXiv:1810.10037). We combine gauge-fixing steps to implement a closed loop in the space of stabilizer codes, returning to the original codespace up to a logical Clifford gate. These measurement-based paths in the space of stabilizer codes can be viewed as shortcuts, or "chutes and ladders", relative to single-qubit Clifford operations and qubit permutations. This yields a machine-interpretable method for constructing dynamical automorphisms and facilitates the search for implementations of desired logical gates. As an example, we implement a logical phase gate via distance-preserving code switching for the seven-qubit code bare code (arXiv:1702.01155), which has no non-trivial logical Clifford gates based on single-qubit Clifford operations and qubit permutations (arXiv:2409.18175).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the ZX-calculus to Floquet and dynamical stabilizer codes by linking measurement-based code switching to gauge fixing. It shows how sequences of gauge-fixing steps can be composed into closed loops in the space of stabilizer codes that return to the original codespace up to a logical Clifford gate; these paths are presented as graphical 'chutes and ladders' shortcuts relative to single-qubit Cliffords and permutations. The central example constructs a logical phase gate on the seven-qubit bare code (which admits no non-trivial logical Cliffords from single-qubit operations and permutations) via distance-preserving code switching, yielding a machine-interpretable method for dynamical automorphisms.

Significance. If the constructions hold, the work supplies a concrete graphical calculus for discovering measurement-based implementations of logical gates that are inaccessible by standard Clifford methods, which is valuable for fault-tolerant architectures on codes with limited transversal gates. The emphasis on machine-interpretable ZX diagrams and the explicit 7-qubit example are strengths that could facilitate automated search for dynamical automorphisms.

major comments (2)
  1. [Section describing closed loops and the 7-qubit example] The central construction (abstract and the section combining gauge-fixing steps into closed loops) asserts that the composition returns to the original codespace up to a unitary logical Clifford; however, no explicit step-by-step computation of the overall codespace map or verification that the composite channel is deterministic and unitary (rather than a probabilistic mixture with residual logical errors) is supplied, leaving the required closure property unverified.
  2. [ZX-calculus extension and 7-qubit example] In the ZX-calculus extension for the seven-qubit bare code (the example implementing the logical phase gate), the manuscript provides no independent diagram reduction or check confirming that the extended rewrite rules preserve the claimed logical action, despite citing that single-qubit Cliffords and permutations are trivial for this code.
minor comments (2)
  1. Notation for the gauge operators and the precise mapping between measurement-based switching and the ZX diagrams could be clarified with an explicit dictionary or table.
  2. Figure captions for the chutes-and-ladders diagrams should explicitly label which segments correspond to which gauge-fixing measurements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address each major comment below, agreeing to make revisions to improve the explicit verification of our claims.

read point-by-point responses

  1. Referee: [Section describing closed loops and the 7-qubit example] The central construction (abstract and the section combining gauge-fixing steps into closed loops) asserts that the composition returns to the original codespace up to a unitary logical Clifford; however, no explicit step-by-step computation of the overall codespace map or verification that the composite channel is deterministic and unitary (rather than a probabilistic mixture with residual logical errors) is supplied, leaving the required closure property unverified.

    Authors: The referee correctly identifies that the current manuscript does not include an explicit step-by-step computation of the composite map. Although the ZX-calculus diagrams are designed to encode and verify the composition via established rewrite rules that preserve stabilizer semantics and ensure unitarity, we acknowledge that a more transparent verification would strengthen the presentation. We will add a detailed calculation of the overall codespace map (using both ZX semantics and an explicit operator representation) for the seven-qubit example in the revised manuscript. revision: yes

  2. Referee: [ZX-calculus extension and 7-qubit example] In the ZX-calculus extension for the seven-qubit bare code (the example implementing the logical phase gate), the manuscript provides no independent diagram reduction or check confirming that the extended rewrite rules preserve the claimed logical action, despite citing that single-qubit Cliffords and permutations are trivial for this code.

    Authors: We agree that an explicit diagram reduction sequence would provide a clearer independent check. The manuscript applies the standard ZX rewrite rules to the extended diagrams for the seven-qubit code, but does not display the full reduction steps. We will include the complete reduction sequence in the revised version, demonstrating that the diagram evaluates to the logical phase gate (and confirming that single-qubit Cliffords and permutations yield only the identity on this code). revision: yes

Circularity Check

0 steps flagged

No circularity: central construction combines external citations into new closed-loop method

full rationale

The derivation relies on the cited connection in arXiv:1810.10037 between measurement-based code switching and gauge fixing, plus standard facts about the seven-qubit code from arXiv:1702.01155 and the absence of certain gates from arXiv:2409.18175. These are external references whose content is not reproduced or redefined inside the paper. The novel step—composing gauge-fixing operations into closed paths that realize logical Cliffords, then rendering them in ZX-calculus—is presented as an independent construction rather than a renaming, fit, or self-referential definition. No equation or claim reduces by construction to its own inputs, and no load-bearing uniqueness theorem is imported from the authors' own prior work. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard ZX-calculus rules and prior arXiv papers on code switching and the seven-qubit code; no new free parameters or entities are introduced in the abstract.

axioms (1)
  • domain assumption ZX-calculus rules can be extended to dynamical stabilizer codes via gauge fixing and code switching.
    The paper assumes the graphical language applies to the new dynamical setting based on the cited connection.

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Reference graph

Works this paper leans on

115 extracted references · 22 canonical work pages · 4 internal anchors

  1. [1]

    and [55, 67]. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. MV was supported in part by Plan France 2030 through the project ANR-22- PETQ0006. SZB was supported b...

    2030

  2. [2]

    Vuillot, L

    C. Vuillot, L. Lao, B. Criger, C. García Almudéver, K. Bertels, and B. M. Terhal, Code deformation and lat- tice surgery are gauge fixing, New Journal of Physics21, 033028 (2019)

    2019

  3. [3]

    M. Li, M. Gutiérrez, S. E. David, A. Hernandez, and K. R. Brown, Fault tolerance with bare ancillary qubits for a [[7,1,3]] code, Phys. Rev. A96, 032341 (2017)

    2017

  4. [4]

    Sayginel, S

    H. Sayginel, S. Koutsioumpas, M. Webster, A. Rajput, and D. E. Browne, Fault-tolerant logical clifford gates from code automorphisms, PRX Quantum6, 030343 (2025)

    2025

  5. [5]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information, 10th ed. (Cambridge Univer- sity Press, Cambridge ; New York, 2010)

    2010

  6. [6]

    N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Ex- tending the lifetime of a quantum bit with error correc- tion in superconducting circuits, Nature536, 441 (2016)

    2016

  7. [7]

    Leroux, C

    S.Krinner, N.Lacroix, A.Remm, A.DiPaolo, E.Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Her- rmann, G.J.Norris, C.K.Andersen, M.Müller, A.Blais, C. Eichler, and A. Wallraff, Realizing repeated quantum error correction in a distance-three surface code, Nature 605, 669 (2022)

    2022

  8. [8]

    Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature614, 676 (2023)

    2023

  9. [9]

    Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature638, 920 (2025)

    2025

  10. [10]

    Lacroix, A

    N. Lacroix, A. Bourassa, F. J. H. Heras,et al., Scaling and logic in the colour code on a superconducting quan- tum processor, Nature645, 614 (2025)

    2025

  11. [11]

    Ryan-Anderson, J

    C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernogu- zov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz, Realization of Real-Time Fault-Tolerant Quantum Error Correction, Physical Re- view X11, 041058 (2021)

    2021

  12. [12]

    Postler, F

    L. Postler, F. Butt, I. Pogorelov, C. D. Marciniak, S. Heußen, R. Blatt, P. Schindler, M. Rispler, M. Müller, and T. Monz, Demonstration of fault-tolerant steane quantum error correction, PRX Quantum5, 030326 (2024)

    2024

  13. [13]

    Demonstration of logical qubits and repeated error correction with better-than-physical error rates

    A. Paetznick, M. P. da Silva, C. Ryan-Anderson, J. M. Bello-Rivas, J. P. C. III, A. Chernoguzov, J. M. Dreil- ing, C. Foltz, F. Frachon, J. P. Gaebler, T. M. Gatter- man, L. Grans-Samuelsson, D. Gresh, D. Hayes, N. He- witt, C. Holliman, C. V. Horst, J. Johansen, D. Luc- chetti, Y.Matsuoka, M.Mills, S.A.Moses, B.Neyenhuis, A. Paz, J. Pino, P. Siegfried, A...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2404.02280 2024

  14. [14]

    B. W. Reichardt, D. Aasen, R. Chao, A. Chernogu- zov, W. van Dam, J. P. Gaebler, D. Gresh, D. Luc- chetti, M. Mills, S. A. Moses, B. Neyenhuis, A. Paet- znick, A. Paz, P. E. Siegfried, M. P. da Silva, K. M. Svore, Z. Wang, and M. Zanner, Demonstration of quan- tum computation and error correction with a tesseract code, arXiv preprint 10.48550/arXiv.2409.0...

    work page doi:10.48550/arxiv.2409.04628 2024

  15. [15]

    Ryan-Anderson, N

    C. Ryan-Anderson, N. C. Brown, C. H. Baldwin, J. M. Dreiling, C. Foltz, J. P. Gaebler, T. M. Gatterman, N. Hewitt, C. Holliman, C. V. Horst, J. Johansen, D. Lucchetti, T. Mengle, M. Matheny, Y. Matsuoka, K. Mayer, M. Mills, S. A. Moses, B. Neyenhuis, J. Pino, P. Siegfried, R. P. Stutz, J. Walker, and D. Hayes, High-fidelity and Fault-tolerant Teleportatio...

    work page doi:10.48550/arxiv.2404.16728 2024

  16. [16]

    J. A. Muniz, D. Crow, H. Kim, J. M. Kindem, W. B. Cairncross, A. Ryou, T. C. Bohdanowicz, C.-A. Chen, Y.Ji, A.M.W.Jones, E.Megidish, C.Nishiguchi, M.Ur- banek, L. Wadleigh, T. Wilkason, D. Aasen, K. Barnes, J. M. Bello-Rivas, I. Bloomfield, G. Booth, A. Brown, M.O.Brown, K.Cassella, G.Cowan, J.Epstein, M.Feld- kamp, C. Griger, Y. Hassan, A. Heinz, E. Halp...

    2025

  17. [17]

    doi:10.48550/arXiv.2506.20661 , url =

    D. Bluvstein, A. A. Geim, S. H. Li, S. J. Evered, J. P. B. Ataides, G. Baranes, A. Gu, T. Manovitz, M. Xu, M. Kalinowski, S. Majidy, C. Kokail, N. Maskara, E. C. Trapp, L. M. Stewart, S. Hollerith, H. Zhou, M. J. Gullans, S. F. Yelin, M. Greiner, V. Vuletic, M. Cain, and M. D. Lukin, Architectural mechanisms of a uni- versal fault-tolerant quantum compute...

    work page doi:10.48550/arxiv.2506.20661 2025

  18. [18]

    Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D

    D. Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D. thesis, Caltech (1997)

    1997

  19. [19]

    Eastin and E

    B. Eastin and E. Knill, Restrictions on transversal en- coded quantum gate sets, Phys. Rev. Lett.102, 110502 (2009)

    2009

  20. [20]

    Horsman, A

    C. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, Surface code quantum computing by lattice surgery, New Journal of Physics14, 123011 (2012)

    2012

  21. [21]

    D.Litinski,AGameofSurfaceCodes: Large-ScaleQuan- tum Computing with Lattice Surgery, Quantum3, 128 (2019)

    2019

  22. [22]

    L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, Low-overhead fault-tolerant quantum computing using long-range connectivity, Science Advances8, eabn1717 (2022)

    2022

  23. [23]

    B. Ide, M. G. Gowda, P. J. Nadkarni, and G. Dauphi- nais, Fault-Tolerant Logical Measurements via Homolog- ical Measurement, Physical Review X15, 021088 (2025)

    2025

  24. [24]

    Swaroop, T

    E. Swaroop, T. Jochym-O’Connor, and T. J. Yoder, Uni- versal adapters between quantum LDPC codes, arXiv preprint 10.48550/arXiv.2410.03628 (2024)

    work page doi:10.48550/arxiv.2410.03628 2024

  25. [25]

    Z. He, A. Cowtan, D. J. Williamson, and T. J. Yoder, Ex- tractors: QLDPC Architectures for Efficient Pauli-Based Computation, arXiv preprint 10.48550/arXiv.2503.10390 (2025)

    work page doi:10.48550/arxiv.2503.10390 2025

  26. [26]

    Knill,Fault-tolerant postselected quantum computation: Schemes,arXiv [quant-ph] (2004), 10.48550/arXiv.quant- ph/0402171, quant-ph/0402171 [quant-ph]

    E. Knill, Fault-Tolerant Postselected Quantum Compu- tation: Schemes, arXiv preprint 10.48550/arXiv.quant- ph/0402171 (2004)

    work page doi:10.48550/arxiv.quant- 2004

  27. [27]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal Clifford gates and noisy ancillas, Physical Review A71, 022316 (2005)

    2005

  28. [28]

    Bravyi and J

    S. Bravyi and J. Haah, Magic-state distillation with low overhead, Phys. Rev. A86, 052329 (2012)

    2012

  29. [29]

    Haah and M

    J. Haah and M. B. Hastings, Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates, Quantum 2, 71 (2018)

    2018

  30. [30]

    Litinski, Magic State Distillation: Not as Costly as You Think, Quantum3, 205 (2019)

    D. Litinski, Magic State Distillation: Not as Costly as You Think, Quantum3, 205 (2019)

    2019

  31. [31]

    Paetznick and B

    A. Paetznick and B. W. Reichardt, Universal fault- tolerant quantum computation with only transversal gates and error correction, Physical Review Letters111, 090505 (2013)

    2013

  32. [32]

    H.Bombín,Gaugecolorcodes: Optimaltransversalgates and gauge fixing in topological stabilizer codes, New Journal of Physics17, 083002 (2015)

    2015

  33. [33]

    Kubica and M

    A. Kubica and M. E. Beverland, Universal transversal gates with color codes: A simplified approach, Phys. Rev. A91, 032330 (2015)

    2015

  34. [34]

    Bombín, Dimensional jump in quantum error correc- tion, New Journal of Physics18, 043038 (2016)

    H. Bombín, Dimensional jump in quantum error correc- tion, New Journal of Physics18, 043038 (2016)

    2016

  35. [35]

    M. E. Beverland, A. Kubica, and K. M. Svore, Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes, PRX Quantum2, 020341 (2021)

    2021

  36. [36]

    M. B. Hastings and J. Haah, Dynamically generated log- ical qubits, Quantum5, 564 (2021)

    2021

  37. [37]

    Davydova, N

    M. Davydova, N. Tantivasadakarn, and S. Balasubra- manian, Floquet codes without parent subsystem codes, PRX Quantum4, 020341 (2023)

    2023

  38. [38]

    M. S. Kesselring, J. C. Magdalena de la Fuente, F. Thom- sen, J. Eisert, S. D. Bartlett, and B. J. Brown, Anyon condensation and the color code, PRX Quantum5, 010342 (2024)

    2024

  39. [39]

    Higgott and N

    O. Higgott and N. P. Breuckmann, Constructions and performance of hyperbolic and semi-hyperbolic floquet codes, PRX Quantum5, 040327 (2024)

    2024

  40. [40]

    Bombín, D

    H. Bombín, D. Litinski, N. Nickerson, F. Pastawski, and S. Roberts, Unifying flavors of fault tolerance with the ZX calculus, Quantum8, 1379 (2024)

    2024

  41. [41]

    Davydova, N

    M. Davydova, N. Tantivasadakarn, S. Balasubramanian, and D. Aasen, Quantum computation from dynamic au- tomorphism codes, Quantum8, 1448 (2024)

    2024

  42. [42]

    Coecke and R

    B. Coecke and R. Duncan, Interacting Quantum Ob- servables: Categorical Algebra and Diagrammatics, New Journal of Physics13, 043016 (2011)

    2011

  43. [43]

    Kissinger and J

    A. Kissinger and J. van de Wetering,Picturing Quan- tum Software: An Introduction to the ZX-Calculus and Quantum Compilation(Book preprint, 2024)

    2024

  44. [44]

    Townsend-Teague, J

    A. Townsend-Teague, J. M. d. l. Fuente, and M. Kessel- ring, Floquetifying the Colour Code, Electronic Proceed- ings in Theoretical Computer Science384, 265 (2023)

    2023

  45. [45]

    Rodatz, B

    B. Rodatz, B. Poór, and A. Kissinger, Floquetifying stabiliser codes with distance-preserving rewrites, arXiv 7 preprint 10.48550/arXiv.2410.17240 (2024)

    work page doi:10.48550/arxiv.2410.17240 2024

  46. [46]

    Jacoby, A

    S. Jacoby, A. Retzker, and F. Pastawski, Stairway Codes: Floquetifying Bivariate Bicycle Codes and Beyond, arXiv preprint 10.48550/arXiv.2603.00228 (2026)

    work page doi:10.48550/arxiv.2603.00228 2026

  47. [47]

    A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. Sloane, Quantum error correction and orthogonal geom- etry, Physical Review Letters78, 405 (1997)

    1997

  48. [48]

    A.Kissinger,Phase-freeZXdiagramsareCSScodes(...or how to graphically grok the surface code), arXiv preprint 10.48550/arXiv.2204.14038 (2022)

    work page doi:10.48550/arxiv.2204.14038 2022

  49. [49]

    A. B. Khesin, J. Z. Lu, and P. W. Shor, Universal graph representation of stabilizer codes, arXiv preprint 10.48550/arXiv.2411.14448 (2025)

    work page doi:10.48550/arxiv.2411.14448 2025

  50. [50]

    Z. Wu, S. Cheng, and B. Zeng, A ZX-Calculus Approach for the Construction of Graph Codes, arXiv preprint 10.48550/arXiv.2304.08363 (2024)

    work page doi:10.48550/arxiv.2304.08363 2024

  51. [51]

    Van den Nest, J

    M. Van den Nest, J. Dehaene,and B. De Moor, Graphical description of the action of local Clifford transformations on graph states, Physical Review A69, 022316 (2004)

    2004

  52. [52]

    Duncan and S

    R. Duncan and S. Perdrix, Graph States and the Neces- sity of Euler Decomposition, inMathematical Theory and Computational Practice(Springer, 2009) pp. 167–177

    2009

  53. [53]

    (2) should be √ X −1 = X √ X(see Theorem 3 of the SM)

    Strictly speaking, the √ Xin Eq. (2) should be √ X −1 = X √ X(see Theorem 3 of the SM). Note however that Pauli operators are transversal gates, and thus come for free from a fault-tolerance perspective. As such we may work in the symplectic representation (see the SM section A), and from now on identify √ X= √ X −1 and √ Z=√ Z −1

  54. [54]

    Bouchet, Graphic presentations of isotropic systems, Journal of Combinatorial Theory, Series B45, 58 (1988)

    A. Bouchet, Graphic presentations of isotropic systems, Journal of Combinatorial Theory, Series B45, 58 (1988)

    1988

  55. [55]

    Duncan and S

    R. Duncan and S. Perdrix, Pivoting makes the ZX- calculus complete for real stabilizers, Electronic Proceed- ings in Theoretical Computer Science171, 50 (2014)

    2014

  56. [56]

    A. T. Hu and A. B. Khesin, Improved graph formalism for quantum circuit simulation, Physical Review A105, 022432 (2022)

    2022

  57. [57]

    D.Kribs, R.Laflamme,andD.Poulin,Unifiedandgener- alized approach to quantum error correction, Phys. Rev. Lett.94, 180501 (2005)

    2005

  58. [58]

    Kribs, R

    D. Kribs, R. Laflamme, D. Poulin, and M. Lesosky, Op- erator quantum error correction, Quantum Information and Computation6, 382 (2006)

    2006

  59. [59]

    Poulin, Stabilizer Formalism for Operator Quantum Error Correction, Physical Review Letters95, 230504 (2005)

    D. Poulin, Stabilizer Formalism for Operator Quantum Error Correction, Physical Review Letters95, 230504 (2005)

    2005

  60. [60]

    Backens, The ZX-calculus is complete for stabilizer quantum mechanics, New Journal of Physics16, 093021 (2014)

    M. Backens, The ZX-calculus is complete for stabilizer quantum mechanics, New Journal of Physics16, 093021 (2014)

    2014

  61. [61]

    In the normal form it is impossible for all wires to have an Hgate as some will necessarily be cancelled by pivoting

  62. [62]

    Aasen, J

    D. Aasen, J. Haah, Z. Li, and R. S. K. Mong, Measure- ment Quantum Cellular Automata and Anomalies in Flo- quet Codes, arXiv preprint 10.48550/arXiv.2304.01277 (2023)

    work page doi:10.48550/arxiv.2304.01277 2023

  63. [63]

    Transversal switching between generic stabilizer codes

    C. Huang and M. Newman, Transversal switch- ing between generic stabilizer codes, arXiv preprint 10.48550/arXiv.1709.09282 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1709.09282 2018

  64. [64]

    K. R. Colladay and E. J. Mueller, Rewiring Stabilizer Codes, New Journal of Physics20, 083030 (2018)

    2018

  65. [65]

    Cross and D

    A. Cross and D. Vandeth, Small Binary Stabilizer Sub- systemCodes,arXivpreprint10.48550/arXiv.2501.17447 (2025)

    arXiv 2025

  66. [66]

    C. D. Hill, A. G. Fowler, D. S. Wang, and L. C. L. Hollen- berg, Fault-tolerant quantum error correction code con- version, Quantum Info. Comput.13, 439 (2013)

    2013

  67. [67]

    Fault-tolerant conversion between stabilizer codes by Clifford operations

    Y. Hwang, B.-S. Choi, Y.-c. Ko, and J. Heo, Fault- tolerant conversion between stabilizer codes by Clifford operations, arXiv preprint 10.48550/arXiv.1511.02596 (2015)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1511.02596 2015

  68. [68]

    A. B. Khesin and A. Li, Equivalence Classes of Quantum Error-Correcting Codes, arXiv preprint 10.48550/arXiv.2406.12083 (2024)

    work page doi:10.48550/arxiv.2406.12083 2024

  69. [69]

    S. Yu, Q. Chen, and C. Oh, Graphical Quan- tum Error-Correcting Codes, arXiv preprint 10.48550/arXiv.0709.1780 (2007)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.0709.1780 2007

  70. [70]

    Morley-Short, gsc (graph-state compass),https:// github.com/sammorley-short/gsc(2019)

    S. Morley-Short, gsc (graph-state compass),https:// github.com/sammorley-short/gsc(2019)

    2019

  71. [71]

    J. C. Adcock, S. Morley-Short, A. Dahlberg, and J. W. Silverstone, Mapping graph state orbits under local com- plementation, Quantum4, 305 (2020)

    2020

  72. [72]

    Aaronson and D

    S. Aaronson and D. Gottesman, Improved Simulation of Stabilizer Circuits, Physical Review A70, 052328 (2004)

    2004

  73. [73]

    With mild abuse of notation, we will occasionally also identify single qubit Cliffords as local Cliffords,LC = Cl(1)

  74. [74]

    Duncan, A

    R. Duncan, A. Kissinger, S. Perdrix, and J. v. d. Weter- ing, Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus, Quantum4, 279 (2020)

    2020

  75. [75]

    StrictlyspeakingitsufficestonotethatanyCliffordisom- etry (as a linear map) may be written as ZX-diagram in the Clifford fragment, but we include this step for ease of the reader

  76. [76]

    M. B. Elliott, B. Eastin, and C. M. Caves, Graphical de- scription of the action of Clifford operators on stabilizer states, Physical Review A77, 042307 (2008)

    2008

  77. [77]

    Gross, Hudson’s theorem for finite-dimensional quan- tum systems, Journal of Mathematical Physics47, 122107 (2006)

    D. Gross, Hudson’s theorem for finite-dimensional quan- tum systems, Journal of Mathematical Physics47, 122107 (2006)

    2006

  78. [78]

    J.DehaeneandB.D.Moor,TheCliffordgroup, stabilizer states, and linear and quadratic operations over GF(2), Physical Review A68, 042318 (2003)

    2003

  79. [79]

    Poór,A unique normal form for prime-dimensional qudit Clifford ZX-calculus, Master’s thesis, University of Oxford (2022)

    B. Poór,A unique normal form for prime-dimensional qudit Clifford ZX-calculus, Master’s thesis, University of Oxford (2022)

    2022

  80. [80]

    Since we work in the symplectic representation, the linear constraint will be always homogoneous

Showing first 80 references.