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A finite ZX diagram with one delay generator completely captures infinite translation-invariant stabilizer processes and reduces them to a unique normal form.

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2026-07-11 22:17 UTC pith:7NZCTD2P

load-bearing objection Solid, self-contained completeness theorem for a natural ZX extension to infinite translation-invariant stabilizer processes; the math checks out.

arxiv 2607.04015 v1 pith:7NZCTD2P submitted 2026-07-04 quant-ph cs.LOmath.CTmath.SG

The Delayed Stabilizer ZX-Calculus

Cole Comfort , Giovanni de Felice This is my paper
classification quant-ph cs.LOmath.CTmath.SG MSC 18M3081P6894B10 PACS 03.67.Pp03.67.Lx
keywords ZX-calculusstabilizer codesquantum convolutional codesdelay generatorgenerating tableauxtranslation-invariant processesgraph statescompleteness
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Stabilizer error-correcting codes and related processes are often defined by a single local pattern repeated forever in space or time. Ordinary ZX diagrams can only draw finite pieces of those patterns, so the infinite symmetry is lost. This paper adds one new generator—the delay—that simply passes data from one time step to the next. The resulting finite diagrams are given two equivalent meanings: one as observational equivalence classes of finite quantum channels whose common information content is an infinite stabilizer group, and one as finite tableaux of rational generating functions whose geometric series recover that group. The authors supply a complete equational theory whose rewrites (generalised Euler rules, colour change, local complementation and pivoting) reduce every diagram to a unique normal form. The result is a sound, universal and complete graphical calculus that lets one reason about infinite translation-invariant stabilizer systems by rewriting finite pictures.

Core claim

The delayed stabilizer ZX-calculus is sound, universal and complete for the generating-tableau semantics of shifted affine Lagrangian relations: every diagram rewrites to a unique reduced AP-form whose data are exactly the canonical generating tableau of the infinite stabilizer group it represents.

What carries the argument

The delay generator δ, interpreted either as a one-step shift on sequences of channels or as multiplication by a formal variable; together with generating tableaux of rational functions and the Schur-complementation rewrite that implements generalised local complementation/pivoting.

Load-bearing premise

Composition of generating tableaux only approximates the composition of their infinite expansions; equality of infinite behaviours therefore rests on observational equivalence of finite unrollings rather than strict channel equality.

What would settle it

Exhibit two delayed ZX diagrams that reduce to distinct reduced AP-forms yet induce the same infinite stabilizer group under geometric-series expansion, or a pair of diagrams that are observationally inequivalent but share the same generating tableau.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any translation-invariant stabilizer process (lattice codes, convolutional codes, infinite graph states) can be drawn as a finite delayed diagram and rewritten to a unique normal form.
  • Stabilizer extraction and local-complementation arguments for infinite codes reduce to finite rational-function arithmetic and graph rewrites.
  • The same normal-form procedure yields a decision procedure for equality of infinite stabilizer groups of delayed diagrams.
  • Catastrophic encoders appear concretely as cases where the geometric-series map is only oplax, giving a diagrammatic test for finite-depth invertibility.

Where Pith is reading between the lines

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

  • The same delay-plus-generating-function pattern should lift to other complete ZX fragments (e.g., Clifford+T) once an appropriate notion of rational phase is supplied.
  • Observational equivalence of unrollings may give a practical finite-window test for whether two convolutional codes have identical distance spectra.
  • The calculus supplies a candidate syntax for automated compilation of infinite-resource MBQC patterns into finite delayed circuits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper introduces the delayed stabilizer ZX-calculus (δZX), extending the odd-prime-dimensional stabilizer ZX-calculus by a single delay generator that encodes translation-invariant infinite processes (lattice codes, convolutional codes, infinite graph states). It supplies two semantics: (i) observational equivalence classes of monotone sequences of completely positive maps / infinite stabilizer groups obtained by unrolling, and (ii) generating tableaux—shifted affine Lagrangian relations over the rational function field F_p(δ)—from which the infinite stabilizers are recovered by geometric series. A complete equational theory is given (generalised spiders, Euler decomposition, colour change), and every diagram is reduced via scalable notation and Schur-complementation (generalised local complementation/pivoting) to a unique reduced AP-form that matches the canonical generating tableau. Theorem 9 establishes that the interpretation into sALR_fd is a †-compact-closed equivalence, hence soundness, universality and completeness for the generating-tableau semantics.

Significance. The work fills a genuine gap: ordinary ZX only describes finite truncations, while the literature on topological and convolutional codes is full of infinite, translation-invariant patterns. By giving a finite graphical language with a complete equational theory and a unique normal form, the paper reduces questions about infinite stabilizer processes to finite rewriting and graph theory. The connection to Haah’s polynomial methods, Wilde–Brun generating tableaux, and the authors’ own observational-behaviour framework is cleanly made. Completeness is proved by an explicit normal-form algorithm (graph-like → AP-form → reduced AP-form) backed by detailed inductive lemmas in the appendices; the construction is therefore machine-checkable in principle and immediately usable for reasoning about quantum convolutional codes and lattice foliations. The honest treatment of oplax composition under geometric expansion (catastrophic encoders) is a strength rather than a defect.

minor comments (6)
  1. The relationship between the two semantics (Γ(Ext(D)) ⊆ StabGrp(D), Proposition 8) is stated correctly but could be flagged more prominently in the introduction and contributions list, so that a reader interested only in infinite channels does not over-read the completeness claim.
  2. Section 6.1 (Definitions 28–32): the case distinction used to define generalised spiders is well-defined (Lemma 13), yet a short forward pointer to the self-conjugate field F_p(δ+δ^{-1}) (Proposition 9) would help the reader before the axioms are stated.
  3. Figure 2 and Axiom 5: the generalised Euler decomposition is stated for self-conjugate labels; a one-line remark that the ordinary ZX Euler rule is recovered when the label is constant would improve readability for ZX specialists.
  4. Example 12 (delayed controlled-X): the infinite propagation of a Pauli-X is illuminating; adding a brief cross-reference to the catastrophic-encoder discussion of Example 8 would tighten the narrative.
  5. Typographical: occasional missing spaces around δ-powers and a few long displayed equations that break across columns in the arXiv rendering; these are purely cosmetic.
  6. References: the concurrent work on observational behaviours (Comfort–de Felice 2026) is cited; once published, a stable bibliographic entry would be preferable.

Circularity Check

1 steps flagged

No significant circularity: completeness is a standard normal-form reduction to an independently defined generating tableau; self-citations supply black-box finite ZX results and the observational framework, not the target equivalence.

specific steps
  1. self citation load bearing [Section 4.1, Proposition 4 and surrounding text; also Theorem 7 / Corollary 2]
    "It follows from essentially the same argument as that of Comfort and de Felice [14, Lem. 6.7], that this yields functorial semantics for stateful ZX-diagrams: Proposition 4. … Theorem 7. There is a faithful †-compact closed discard functor Lim: Obs(ALR_fd) o AR_∞ … [14, Cor. 5.6]"

    The observational-behaviour semantics and the infinite-stabilizer-group limit are justified by direct appeal to the authors’ concurrent/prior paper [14]. This is load-bearing for the first semantics (Obs and StabGrp) but is not used in the proof of the main completeness result (Theorem 9) for the generating-tableau semantics; hence only a minor, non-central circularity flag.

full rationale

The load-bearing claim (Theorem 9) is that the interpretation of δZX into sALR_fd is a †-CC equivalence. Soundness of the axioms is direct verification against the generating-tableau semantics (Section 6.2). Universality and completeness follow from an explicit rewrite sequence (Lemmas 7, Props. 12–14) that puts every diagram into reduced AP-form, whose data are precisely the unique canonical generating tableau of Proposition 6 (proved by elementary linear algebra over F_p(δ) in Appendix B). The normal form is therefore not defined in terms of the rewrite rules; the rules are shown to reach the independently unique tableau. Composition under geometric-series expansion is only oplax (Theorem 8), but that fact is internal and does not affect completeness for sALR_fd itself. Self-citations ([7] for finite ZX, [14] for Obs and Lim) are used as black boxes for prior fragments; they do not force the new normal-form uniqueness or the delayed axioms. No fitted parameters, no self-definitional loop, and no uniqueness theorem imported solely by author citation appear in the central chain. Score 1 reflects only the minor, non-load-bearing self-citation for the secondary observational semantics.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 3 invented entities

The paper rests on standard finite-dimensional stabilizer theory, the existing odd-prime ZX completeness results, and the authors’ own prior work on observational behaviours of monoidal processes. The only genuinely new postulates are the delay generator together with its sliding equations and the definition of shifted symplectic form / generating tableaux; everything else is derived or cited.

axioms (5)
  • domain assumption Odd-prime-dimensional stabilizer ZX-calculus is sound, universal and complete for affine Lagrangian relations (Booth–Carette, Poór et al., Booth–Carette–Comfort).
    Used as the base language that is extended by the delay; completeness of δZX inherits from this result.
  • domain assumption Observational equivalence of monotone sequences of completely-positive maps (Comfort–de Felice) correctly captures ‘same information content’.
    Underpins the first semantics and the faithfulness of the infinite-stabilizer-group functor.
  • ad hoc to paper The delay generator satisfies only the sliding equations with spiders (Definition 17).
    The minimal equational theory needed to make unrolling well-defined up to observational equivalence.
  • ad hoc to paper Shifted symplectic form ϖ_n(f,g) = ω_n(f(δ),g(δ^{-1})) correctly encodes translation-invariant commutation of Pauli sequences.
    Definition 24; the entire generating-tableau semantics rests on this sesquilinear form.
  • standard math Standard linear algebra over the field of rational functions F_p(δ) (Gaussian elimination, Hermitian matrices, Schur complements).
    Used throughout the normal-form proofs.
invented entities (3)
  • delay generator δ no independent evidence
    purpose: Finite syntactic device that encodes the infinite translation-invariant repetition of a pattern.
    New generator added to ZX; its interpretation as multiplication by the indeterminate δ is the bridge to generating functions.
  • generating tableau (shifted affine Lagrangian relation over F_p(δ)) no independent evidence
    purpose: Finite matrix of rational functions that encodes an infinite stabilizer group via geometric series.
    Generalises ordinary stabilizer tableaux; uniqueness of canonical form (Proposition 6) is proved from scratch.
  • observational behaviours Obs(C) of monotone sequences no independent evidence
    purpose: Equivalence classes of finite unrollings that identify processes with the same infinite information content.
    Taken from the authors’ concurrent work but specialised here to stabilizer ZX; used to obtain a genuine Hilbert-space semantics.

pith-pipeline@v1.1.0-grok45 · 41435 in / 2659 out tokens · 27058 ms · 2026-07-11T22:17:53.106164+00:00 · methodology

0 comments
read the original abstract

Many stabilizer quantum error-correcting codes are built from a finite pattern repeated across space or time, such as lattice codes, translation-invariant graph states, and quantum convolutional codes. Ordinary stabilizer ZX-diagrams capture only finite truncations of such systems, obscuring the repeated structure that defines them. We introduce the delayed stabilizer ZX-calculus, a finite graphical language for these infinite, translation-invariant processes. It extends the odd-prime-dimensional stabilizer ZX-calculus with a single new generator, the delay, which feeds data from one time step to the next. We equip the calculus with two semantics. In the first semantics, we interpret the behaviour of a delayed ZX-diagram as an equivalence class of sequences of quantum channels; where two sequences are identified if they have the same information content. We show that the behaviour of a delayed ZX-diagram uniquely determines an infinite stabilizer group. In the second semantics, we interpret the delay as a formal variable, encoding the translation-invariant families of Pauli operators as generating functions. This allows us to represent a delayed ZX-diagram in terms of a tableau of generating functions, from which the infinite stabilizer group can be recovered. Finally, we give a complete axiomatization of the delayed stabilizer ZX-calculus, featuring generalised Euler decomposition and colour change rules. Using generalised forms of local complementation and pivoting, we reduce every diagram to a unique normal form. This establishes soundness, universality, and completeness for the generating tableau semantics.

Figures

Figures reproduced from arXiv: 2607.04015 by Cole Comfort, Giovanni De Felice.

Figure 1
Figure 1. Figure 1: Axioms of ZX, for all a,b,c,d,z ∈ Fp, with z ̸= 0 and permutations ς and τ. From now on, however, we will only interpret stabilizer ZX-diagrams in ALRfd, thereby forgoing the need to keep track of nonzero scalars: t m . . . n . . . a b | := n [ x z ], [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Axioms of the delayed stabilizer ZX-calculus. [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Infinitely unrolled surface code of Example 14. [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Axioms of the Graphical Affine Algebra [7] (GAA), where [PITH_FULL_IMAGE:figures/full_fig_p040_4.png] view at source ↗

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Reference graph

Works this paper leans on

39 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    093021, doi:10.1088/1367-2630/16/9/093021

    Miriam Backens (2014):The ZX-calculus is complete for stabilizer quantum mechanics.New Journal of Physics16(9), p. 093021, doi:10.1088/1367-2630/16/9/093021

    doi:10.1088/1367-2630/16/9/093021 2014

  2. [2]

    Baez & Jason Erbele (2015):Categories in Control.Theory and Applications of Categories30(24), pp

    John C. Baez & Jason Erbele (2015):Categories in Control.Theory and Applications of Categories30(24), pp. 836–881. Available athttp://www.tac.mta.ca/tac/volumes/30/24/30-24.pdf

    2015

  3. [3]

    In: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Lan- guages, POPL ’15, ACM, p

    Filippo Bonchi, Paweł Soboci ´nski & Fabio Zanasi (2015):Full Abstraction for Signal Flow Graphs. In: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Lan- guages, POPL ’15, ACM, p. 515–526, doi:10.1145/2676726.2676993

    doi:10.1145/2676726.2676993 2015

  4. [4]

    2–29, doi:10.1016/j.ic.2016.03.002

    Filippo Bonchi, Paweł Soboci ´nski & Fabio Zanasi (2017):The Calculus of Signal Flow Diagrams I: Linear relations on streams.Information and Computation252, p. 2–29, doi:10.1016/j.ic.2016.03.002. Available at https://eprints.soton.ac.uk/396532/

    doi:10.1016/j.ic.2016.03.002 2017

  5. [5]

    Filippo Bonchi, Paweł Soboci´nski & Fabio Zanasi (2021):A Survey of Compositional Signal Flow Theory, p. 29–56. Springer International Publishing, doi:10.1007/978-3-030-81701-5 2. Available athttps://inria. hal.science/hal-03325995v1/document

    work page doi:10.1007/978-3-030-81701-5 2021

  6. [6]

    Booth & Titouan Carette (2022):Complete ZX-Calculi for the Stabilizer Fragment in Odd Prime Dimensions

    Robert I. Booth & Titouan Carette (2022):Complete ZX-Calculi for the Stabilizer Fragment in Odd Prime Dimensions. In Stefan Szeider, Robert Ganian & Alexandra Silva, editors:47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022),Leibniz International Proceedings in In- formatics (LIPIcs)241, Schloss Dagstuhl – Leibniz-Zen...

    doi:10.4230/lipics.mfcs.2022.24 2022

  7. [7]

    Booth, Titouan Carette & Cole Comfort (2024):Graphical Symplectic Algebra

    Robert I. Booth, Titouan Carette & Cole Comfort (2024):Graphical Symplectic Algebra. arXiv:2401.07914v3. Accepted to FSCD 2026

    Pith/arXiv arXiv 2024

  8. [8]

    Booth & Cole Comfort:Denotational semantics for stabilizer quantum programs

    Robert I. Booth & Cole Comfort:Denotational semantics for stabilizer quantum programs. arXiv:2511.22734v1. Accepted to FSCD 2026

    arXiv 2026

  9. [9]

    1–28, doi:10.1145/3464693

    Titouan Carette, Emmanuel Jeandel, Simon Perdrix & Renaud Vilmart (2021):Completeness of Graphical Languages for Mixed State Quantum Mechanics.ACM Transactions on Quantum Computing2(4), p. 1–28, doi:10.1145/3464693

    work page doi:10.1145/3464693 2021

  10. [10]

    Titouan Carette, Marc de Visme & Simon Perdrix (2021):Graphical language with delayed trace: pictur- ing quantum computing with finite memory. In:Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’21, Association for Computing Machinery, New York, NY , USA, doi:10.1109/LICS52264.2021.9470553. arXiv:2102.03133

    doi:10.1109/lics52264.2021.9470553 2021

  11. [11]

    76–82, doi:10.1103/physreva.56.76

    Richard Cleve & Daniel Gottesman (1997):Efficient computations of encodings for quantum error correc- tion.Physical Review A56(1), p. 76–82, doi:10.1103/physreva.56.76. arXiv:quant-ph/9607030

    Pith/arXiv arXiv doi:10.1103/physreva.56.76 1997

  12. [12]

    043016, doi:10.1088/1367-2630/13/4/043016

    Bob Coecke & Ross Duncan (2011):Interacting quantum observables: categorical algebra and diagram- matics.New Journal of Physics13(4), p. 043016, doi:10.1088/1367-2630/13/4/043016

    doi:10.1088/1367-2630/13/4/043016 2011

  13. [13]

    Cambridge University Press, doi:10.1017/9781316219317

    Bob Coecke & Aleks Kissinger (2017):Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, doi:10.1017/9781316219317

    doi:10.1017/9781316219317 2017

  14. [14]

    Cole Comfort & Giovanni de Felice (2026):Finite Observations, Infinite Behaviour: Bicategorical Semantics for Stateful Monoidal Processes

    2026

  15. [15]

    In:Electronic Proceedings in Theoretical Computer Science, 372, EPTCS, pp

    Cole Comfort & Aleks Kissinger (2022):A Graphical Calculus for Lagrangian Relations. In:Electronic Proceedings in Theoretical Computer Science, 372, EPTCS, pp. 338–351, doi:10.4204/EPTCS.372.24

    doi:10.4204/eptcs.372.24 2022

  16. [16]

    4452–4505, doi:10.1063/1.1499754

    Eric Dennis, Alexei Kitaev, Andrew Landahl & John Preskill (2002):Topological quantum memory.Journal of Mathematical Physics43(9), p. 4452–4505, doi:10.1063/1.1499754. arXiv:quant-ph/0110143

    Pith/arXiv arXiv doi:10.1063/1.1499754 2002

  17. [17]

    Daniel Gottesman (1997):Stabilizer Codes and Quantum Error Correction. Ph.D. thesis, Caltech. arXiv:quant-ph/9705052

    Pith/arXiv arXiv 1997

  18. [18]

    arXiv:0602001

    David Gross (2006):Hudson’s theorem for finite-dimensional quantum systems.Journal of Mathematical Physics47(12), doi:10.1063/1.2393152. arXiv:0602001. C. Comfort and G. de Felice31

    doi:10.1063/1.2393152 2006

  19. [19]

    351–399, doi:10.1007/s00220-013-1810-2

    Jeongwan Haah (2013):Commuting Pauli Hamiltonians as Maps between Free Modules.Communications in Mathematical Physics324(2), p. 351–399, doi:10.1007/s00220-013-1810-2. arXiv:1204.1063

    Pith/arXiv arXiv doi:10.1007/s00220-013-1810-2 2013

  20. [20]

    299, doi:10.15446/recolma.v50n2.62214

    Jeongwan Haah (2017):Algebraic Methods for Quantum Codes on Lattices.Revista Colombiana de Matem´aticas50(2), p. 299, doi:10.15446/recolma.v50n2.62214

    doi:10.15446/recolma.v50n2.62214 2017

  21. [21]

    Journal of Mathematical Physics62(9), doi:10.1063/5.0022185

    Jeongwan Haah (2021):Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D. Journal of Mathematical Physics62(9), doi:10.1063/5.0022185. arXiv:1907.02075

    Pith/arXiv arXiv doi:10.1063/5.0022185 2021

  22. [22]

    David Forney Jr & Saikat Guha (2005):Simple Rate-1/3 Convolutional and Tail-Biting Quantum Error- Correcting Codes

    G. David Forney Jr & Saikat Guha (2005):Simple Rate-1/3 Convolutional and Tail-Biting Quantum Error- Correcting Codes. In:Proceedings. International Symposium on Information Theory, 2005. ISIT 2005., pp. 1028–1032, doi:10.1109/ISIT.2005.1523495. arXiv:quant-ph/0501099

    work page doi:10.1109/isit.2005.1523495 2005

  23. [23]

    Katis, N

    P. Katis, N. Sabadini & R.F.C. Walters (1997):Bicategories of processes.Journal of Pure and Applied Algebra115(2), p. 141–178, doi:10.1016/s0022-4049(96)00012-6

    doi:10.1016/s0022-4049(96)00012-6 1997

  24. [24]

    arXiv:arXiv:2204.14038

    Aleks Kissinger (2022):Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code). arXiv:arXiv:2204.14038

    Pith/arXiv arXiv 2022

  25. [25]

    Preprint

    Aleks Kissinger & John van de Wetering (2024):Picturing Quantum Software: An Introduction to the ZX- Calculus and Quantum Compilation. Preprint. Available athttps://zxcalc.github.io/book

    2024

  26. [26]

    Kitaev (2003):Fault-tolerant quantum computation by anyons.Annals of Physics303(1), p

    A.Yu. Kitaev (2003):Fault-tolerant quantum computation by anyons.Annals of Physics303(1), p. 2–30, doi:10.1016/s0003-4916(02)00018-0. arXiv:quant-ph/9707021

    Pith/arXiv arXiv doi:10.1016/s0003-4916(02)00018-0 2003

  27. [27]

    Werner:Quantum Channels with Memory72(6), p

    Dennis Kretschmann & Reinhard F. Werner:Quantum Channels with Memory72(6), p. 062323. doi:10.1103/PhysRevA.72.062323. arXiv:quant-ph/0502106

    Pith/arXiv arXiv doi:10.1103/physreva.72.062323

  28. [28]

    Neretin (2011):Lectures on Gaussian Integral Operators and Classical Groups.EMS Series of Lectures in Mathematics91, European Mathematical Society

    Yurii A. Neretin (2011):Lectures on Gaussian Integral Operators and Classical Groups.EMS Series of Lectures in Mathematics91, European Mathematical Society. Available athttps://www.mat.univie. ac.at/%7Eneretin/lectures/chapter9.ps

    2011

  29. [29]

    Briegel:Universal resources for measurement-based quantum computation97(15), p

    Maarten Van den Nest, Akimasa Miyake, Wolfgang D ¨ur & Hans J. Briegel:Universal resources for measurement-based quantum computation97(15), p. 150504. doi:10.1103/PhysRevLett.97.150504. arXiv:quant-ph/0604010

    Pith/arXiv arXiv doi:10.1103/physrevlett.97.150504

  30. [30]

    Nielsen & Isaac L

    Michael A. Nielsen & Isaac L. Chuang (2012):Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, doi:10.1017/cbo9780511976667

    doi:10.1017/cbo9780511976667 2012

  31. [31]

    arXiv:0304189

    Harold Ollivier & Jean-Pierre Tillich (2003):Description of a Quantum Convolutional Code.Physical Review Letters91(17), doi:10.1103/physrevlett.91.177902. arXiv:0304189

    doi:10.1103/physrevlett.91.177902 2003

  32. [32]

    2776–2798, doi:10.1109/tit.2009.2018339

    David Poulin, Jean-Pierre Tillich & Harold Ollivier (2009):Quantum Serial Turbo Codes.IEEE Transactions on Information Theory55(6), p. 2776–2798, doi:10.1109/tit.2009.2018339. arXiv:0712.2888

    doi:10.1109/tit.2009.2018339 2009

  33. [33]

    Boldizs ´ar Po ´or, Robert I. Booth, Titouan Carette, John van de Wetering & Lia Yeh (2023):The Qupit Sta- bilizer ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification.Electronic Proceedings in Theoretical Computer Science384, p. 220–264, doi:10.4204/eptcs.384.13

    doi:10.4204/eptcs.384.13 2023

  34. [34]

    Peter Selinger (2004):Towards a semantics for higher-order quantum computation. pp. 127–143. Available athttps://mathstat.dal.ca/ ~selinger/qpl2004/PDFS/09Selinger.pdf

    2004

  35. [35]

    category

    Alan Weinstein (1982):The symplectic “category”, p. 45–51. Springer Berlin Heidelberg, doi:10.1007/bfb0092426

    doi:10.1007/bfb0092426 1982

  36. [36]

    arXiv:arXiv:2012.13966

    John van de Wetering (2020):ZX-calculus for the working quantum computer scientist. arXiv:arXiv:2012.13966

    Pith/arXiv arXiv 2020

  37. [37]

    Wilde (2008):Quantum coding with entanglement

    Mark M. Wilde (2008):Quantum coding with entanglement. Ph.D. thesis, University of Southern California. arXiv:0806.4214

    Pith/arXiv arXiv 2008

  38. [38]

    Entanglement-Assisted Quantum Convolutional Coding

    Mark M. Wilde & Todd A. Brun (2010):Entanglement-assisted quantum convolutional coding.Physical Review A81(4), doi:10.1103/physreva.81.042333. arXiv:0712.2223

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreva.81.042333 2010

  39. [39]

    Wilde, Hari Krovi & Todd A

    Mark M. Wilde, Hari Krovi & Todd A. Brun (2010):Convolutional entanglement distillation. In:2010 IEEE International Symposium on Information Theory, IEEE, p. 2657–2661, doi:10.1109/isit.2010.5513666. arXiv:0708.3699. 32The Delayed Stabilizer ZX-Calculus A Proofs of Section 3 Proposition 2.Proj(CPM)is a discard bicategory with respect to the trace and thep...

    work page doi:10.1109/isit.2010.5513666 2010