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REVIEW 3 major objections 5 minor 39 references

Color-code logical circuits can be compiled automatically into valid spacetime block layouts that beat surface-code volume on standard benchmarks.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-31 05:15 UTC pith:OF62HHKB

load-bearing objection Real first automated color-code logical compiler with topology-grounded blocks and useful volume wins; correctness is by-construction ZX, not independently certified after routing. the 3 major comments →

arxiv 2607.28504 v1 pith:OF62HHKB submitted 2026-07-30 quant-ph

Spacetime Layout and Logical Compilation of Color Code

Qinjing Yu , Ke Liu This is my paper
classification quant-ph
keywords color codelogical compilationspacetime layoutZX calculuslattice surgeryfusion-region-aware routingClifford+Tfault-tolerant quantum computing
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.

Fault-tolerant quantum computers need more than isolated logical gates; whole algorithms must be turned into coordinated arrangements of logical patches over space and time. This paper builds that missing compilation layer for the two-dimensional color code. It defines spacetime blocks—prisms for patches, pipes for connections, ports for inputs and magic states—together with assembly rules forced by the code’s anyon and domain-wall structure, then links those blocks to ZX diagrams so the layout’s meaning is exact. A code-specific rewrite produces edge-decorated ZX diagrams that fit the color code’s junction limits, and fusion-region-aware routing reuses equivalent geometry to shrink volume. An automated pipeline compiles Clifford+T circuits across nine algorithms into validated layouts whose bounding-box spacetime volume is smaller than reported surface-code baselines.

Core claim

The authors show that universal Clifford+T computations can be compiled, fully automatically, into color-code spacetime layouts that obey the code’s topological assembly rules, preserve the intended logical map via a ZX correspondence, and achieve lower bounding-box spacetime volume than surface-code compilation on the same nine algorithmic benchmarks.

What carries the argument

Edge-decorated ZX diagrams plus fusion-region-aware routing: the former rewrites a circuit into phase-free spiders of degree at most five with Clifford sequences on edges (matching color-code junctions and free transversal Cliffords); the latter embeds whole fusion regions so routes may attach anywhere in a region, reusing geometry and pruning degree-one leaves while preserving semantics.

Load-bearing premise

That matching block-assembly rules and the junction-to-spider Pauli-web map is enough to guarantee a correct logical computation, even though the actual physical syndrome circuits and distance-dependent fault tolerance are left for a later layer and are not checked here.

What would settle it

Compile one of the nine benchmark circuits, extract the resulting layout’s ZX diagram, and check whether it is ZX-equivalent to the input circuit’s diagram and whether every junction obeys the stated colour/Pauli matching and degree rules; any validated layout that fails either check, or a volume comparison that no longer beats the surface-code baseline under the paper’s own metrics, would refute the claim.

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

If this is right

  • Color-code FTQC can move from hand-built primitives to algorithm-scale automated logical synthesis.
  • Degree-5 junctions and free transversal single-qubit Cliffords become systematic volume-reduction levers versus surface-code layouts.
  • The same block/ZX interface can host other search engines or boundary and syndrome-extraction choices without rewriting the logical layer.
  • Compiled layouts supply a stable intermediate representation for joint optimization with downstream physical-circuit compilation.

Where Pith is reading between the lines

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

  • If the macroscopic rules truly separate from microscopic fault tolerance, color-code full-stack stacks can evolve the physical layer independently while keeping the logical compiler fixed.
  • Fusion-region reuse may generalize to other codes whose anyon condensation yields high-degree merges, including some QLDPC families the paper flags.
  • Occupancy well below bounding-box volume on the benchmarks suggests a second compaction pass could close more of the remaining spacetime gap.

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

3 major / 5 minor

Summary. The paper develops a logical compilation framework for the 2D (6.6.6) color code. It defines a spacetime block language (prisms, time/space pipes, ports) from anyon condensation and domain walls, derives assembly rules (connectivity, Pauli/colour matching, degree ≤5 junctions), and establishes a junction–spider correspondence to ZX diagrams (Sec. 2, Appendix A). Compilation proceeds by rewriting circuits into edge-decorated ZX diagrams under those constraints, then embedding them with fusion-region-aware routing (deferred prism basis, leaf pruning, transparent-wall re-anchoring). An automated MCTS-based pipeline compiles Clifford+T circuits; on nine benchmarks it reports assembly-rule-valid layouts and smaller bounding-box spacetime volume than TopoLS surface-code baselines (Table 1), with further gains from fusion-region awareness. Physical syndrome circuits and distance-dependent fault tolerance are left downstream. Concurrent pipe-diagram work by Herzog et al. is distinguished as different in scope and automation.

Significance. Logical compilation for the color code has lagged surface-code architectures despite transversal Cliffords and richer domain-wall structure. A code-derived block language, ZX semantics, and an automated algorithm-scale pipeline that produces validated layouts are a genuine step from isolated primitives toward full-stack color-code FTQC. Fusion-region-aware routing and degree-5 junctions are concrete, reusable ideas; the explicit correctness criterion via ZX equivalence is the right standard; and the planned public code/data release would make the contribution checkable. If the end-to-end semantic claim is secured and the cross-code volume comparison is carefully scoped, the work is a solid reference point for color-code architecture and for compilation methods on other topological/QLDPC codes.

major comments (3)
  1. [Sec. 2.4, Sec. 3.2, Sec. 4, Table 1] Sec. 2.4 states that a compiled layout is correct precisely when its ZX diagram is equivalent to that of the input circuit. Sec. 3.2–4 assert equivalence “by construction” via rewriting, fusion/unfusion, pruning, and wall re-anchoring, and report only that returned layouts pass the Sec. 2.3 assembly-rule check plus volume metrics (Table 1, Fig. 5). After MCTS placement, deferred-basis resolution, leaf pruning, transparent-domain-wall re-anchoring, and multi-block stitching, assembly validity does not by itself discharge the stated ZX criterion. For the nine benchmarks, please either (i) extract the final layout ZX and check equivalence to the input (rewrite transcript, Pauli-web/stabilizer certificate, or small independent checker), or (ii) give an invariant argument that each geometric step preserves the linear map, including deferred-basis resolution and re-anchoring. Without one of th
  2. [Table 1, Sec. 4, Abstract] Table 1 and the abstract/Sec. 4 claim uniformly smaller bounding-box volume than TopoLS. TopoLS compiles to surface-code lattice surgery; this work compiles to color-code prisms/pipes with transversal single-qubit Cliffords and degree-5 junctions. The comparison is informative as a cross-architecture resource snapshot only if the volume units, circuit suite, and normalization (e.g., what one “prism” time step counts vs a surface-code cube, distance scaling, magic-state accounting) are stated explicitly. As written, it is unclear whether the wins measure compilation quality, native color-code advantages, or incomparable units. Please define the volume model side-by-side and separate “fusion vs inplace” (same code; clean ablation) from “this work vs TopoLS” (cross-code; needs caveats).
  3. [Appendix A, Sec. 2.4, Sec. 3.2] Appendix A shows that colour-boundary three-Pauli symmetry forces a conventional identification of space-pipe anyon strings with definite Pauli labels to match ZX spiders, and notes that Y-strings can appear to violate the Pauli web without that choice. Sec. 3.2 then defers prism basis along routes and resolves it when a region claims the prism. The manuscript should state how deferred-basis resolution and the Appendix A identification are kept consistent at every junction (especially after pruning/re-anchoring), and whether an invalid identification can pass the assembly-rule validator. This is part of the semantic bridge, not only a microscopic detail.
minor comments (5)
  1. [Introduction, Sec. 5] Several word-spacing glitches appear in the extracted text (e.g., “Ourframeworkmoves…”, “Duringthepreparation…”, “experismental”). Please proofread the camera-ready source for run-on words and typos.
  2. [Fig. 3, Fig. 6] Fig. 3 and Fig. 6 are central to the junction–spider story; ensure labels (heights, domain-wall types, spider kind) remain legible and that the caption states that X-boundary junctions are analogous.
  3. [Sec. 4] Sec. 4: briefly list MCTS hyperparameters (candidate moves, restarts, reward, seeds) and the deterministic fallback builder’s volume model so Table 1 is reproducible when code is released.
  4. [Sec. 2.2, Table 1] Clarify early that magic-state injection ports are assumed available asynchronously and that factory cost is outside the reported spacetime volume, so Table 1 is layout volume for the algorithm body only.
  5. [Introduction, Acknowledgments] The distinction from Herzog et al. [27] is helpful; a short explicit table (scope, automation, Clifford vs Clifford+T, physical-circuit layer) would make the concurrent-work discussion easier to cite.

Circularity Check

0 steps flagged

No significant circularity: constructive compiler with ZX-preserving rewrites and measured volumes, not predictions forced by inputs.

full rationale

The paper’s load-bearing chain is definitional and constructive rather than predictive. Spacetime blocks, assembly rules, and the junction–spider correspondence (Sec. 2.2–2.4, Appendix A) are derived from color-code anyon condensation and Pauli-web propagation; they are not fitted to the benchmark volumes. Edge decoration and fusion-region-aware routing are designed so that ZX rewrite/fusion/unfusion preserve the linear map “by construction” (Sec. 3.1–3.2, Sec. 4)—standard compiler semantics, not a fitted input renamed as a prediction. Table 1 volumes are empirical outputs of the automated pipeline versus TopoLS baselines, not quantities used to calibrate the method. Concurrent Herzog et al. is acknowledged as independent and non-load-bearing. Correctness is defined as ZX equivalence to the input circuit; asserting preservation via calculus rules is self-contained methodology, not circular reduction of a claimed first-principles result to its own inputs. Any gap between assembly-rule validation and post-routing ZX certificates is a verification/correctness concern, not circularity under this pass.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 3 invented entities

The central claim rests on standard color-code/anyon and ZX background, plus paper-specific block conventions and the assumption that rule-valid block graphs with ZX-equivalent diagrams are the right logical-compilation target while physics is downstream. No numerical fitting to nature; free choices are compiler/search design parameters and layout conventions.

free parameters (3)
  • MCTS and pipeline search hyperparameters (move sets, restarts, rewards, eight random seeds, best-run selection) = best of 8 seeds per circuit/config (Table 1 protocol)
    Embedding quality and reported volumes depend on heuristic search configuration; only high-level reward (negative spacetime volume) and best-of-eight selection are described, not a full hyperparameter schedule.
  • Circuit blocking / layering cut policy (one port per qubit; bisection on failure)
    How circuits are sliced into independently compiled blocks affects layout volume and success; the policy is specified qualitatively and can change outcomes.
  • Space-pipe face convention (colour vs Pauli top/bottom faces) = colour boundaries on space-pipe top/bottom
    Authors choose colour boundaries WLOG for compilation (Sec. 2.2, Appendix A); they claim macroscopic equivalence but note microscopic circuit differences.
axioms (6)
  • domain assumption 2D hexagonal color-code anyon model, boson table, and domain-wall taxonomy (opaque/semi-transparent/transparent) as in the cited color-code literature
    Sec. 2.1 grounds blocks and matching rules in this standard topological structure.
  • standard math ZX calculus rewrite rules preserve the represented linear map; Pauli-web rules characterize phase-0 spiders
    Sec. 2.4 and 3 use ZX equivalence as the semantic correctness criterion for layouts and rewrites.
  • domain assumption Transversal single-qubit Cliffords on the 2D color code may be stacked as free transparent domain walls on time pipes without extra spacetime volume in this representation
    Sec. 2.2 and edge decoration treat H/S sequences as volume-free decorations.
  • domain assumption Non-Clifford resource is asynchronous |T⟩ injection at designated ports; factories and injection fidelity are external
    Sec. 2.2 restricts the layout language to Clifford+T with magic-state ports.
  • ad hoc to paper Assembly rules (connectivity, Pauli matching/overlap, colour matching) plus ZX equivalence suffice to call a layout a correct logical compilation; distance-dependent stabilizers and syndrome circuits are out of scope
    Stated compilation target and validation criterion in Sec. 2–4; physical realization explicitly left downstream, including vs Herzog et al. microscopic compilation.
  • ad hoc to paper Conventional identification of deconfined space-pipe anyon strings with definite Pauli labels is enough to match ZX spiders despite colour-boundary Pauli symmetry
    Appendix A notes an apparent Y-propagation subtlety and resolves it by choosing an identification.
invented entities (3)
  • Color-code spacetime block language (prism, time pipe, space pipe/cubes, ports) with degree-≤5 junctions no independent evidence
    purpose: Intermediate representation capturing patches and operations under color-code domain walls
    Introduced in Sec. 2.2–2.3 as the layout IR; related to concurrent pipe diagrams but with distinct conventions.
  • Edge-decorated ZX diagram for color code no independent evidence
    purpose: Phase-free skeleton with H/S edge decorations and |T⟩ boundary ports matching block constraints
    Sec. 3.1 rewriting target tailored to transversal Cliffords and degree bounds.
  • Fusion-region-aware routing with deferred prism basis no independent evidence
    purpose: Embed edges by attaching anywhere in fusion regions to reduce spacetime volume while preserving ZX semantics
    Sec. 3.2; volume gains in Table 1 are the internal performance evidence only.

pith-pipeline@v1.2.0-daily-grok45 · 19656 in / 4018 out tokens · 79780 ms · 2026-07-31T05:15:20.527401+00:00 · methodology

0 comments
read the original abstract

Fault-tolerant quantum computing requires system-level coordination of logical primitives. Here, we establish a logical compilation framework for the color code, grounded in its topological structure and supporting universal logical operations. Based on its anyon-condensation and domain-wall structure, we introduce a spacetime block-diagram representation capturing logical patches and operations and derive the rules governing block assembly. A correspondence with ZX diagrams further identifies the logical semantics of this representation and enables transformations that preserve the represented computation. Moreover, we develop a code-derived compilation strategy that converts ZX representations of logical computations into valid color-code spacetime layouts. In this strategy, edge-decorated ZX diagrams tailor the logical representation to the color code under the block-assembly constraints, and fusion-region-aware routing exploits semantic equivalence during geometric embedding. We automate the complete logical compilation process and demonstrate successful compilation across a broad range of algorithms. Our work advances color-code architecture from individual primitives to the automated synthesis of logical computations, marking a significant step toward its full-stack quantum computing.

Figures

Figures reproduced from arXiv: 2607.28504 by Ke Liu, Qinjing Yu.

Figure 1
Figure 1. Figure 1: Left: The anyons in the color code. A string of physical [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The building blocks of the spacetime layout of the 2D color code. Left: a prism, with its top [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: All possible z-basis junctions of degree 2, 3, 4, and 5 in the spacetime layout of the 2D [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: Two prisms with deferred basis in the spatial direction, and the corresponding rep [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The result of our compilation pipeline for the 16-qubit VQE circuit. Left: the compiled [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: A degree-5 junction in the spacetime layout of the 2D color code, with domain walls [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

39 extracted references · 13 linked inside Pith

  1. [1]

    Topologicalquantum memory

    EricDennis, AlexeiKitaev, AndrewLandahl, andJohnPreskill. “Topologicalquantum memory”. Journal of Mathematical Physics43, 4452–4505 (2002)

    2002

  2. [2]

    Surface codes: Towards practical large-scale quantum computation

    Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. “Surface codes: Towards practical large-scale quantum computation”. Physical Review A86, 032324 (2012). 16

    2012

  3. [3]

    Sur- face code quantum computing by lattice surgery

    Dominic Horsman, Austin G. Fowler, Simon Devitt, and Rodney Van Meter. “Sur- face code quantum computing by lattice surgery”. New Journal of Physics14, 123011 (2012)

    2012

  4. [4]

    Yokedsurfacecodes

    CraigGidney, MichaelNewman, PeterBrooks, andCodyJones. “Yokedsurfacecodes”. Nature Communications16, 4498 (2025)

    2025

  5. [5]

    Inplace access to the surface code Y basis

    Craig Gidney. “Inplace access to the surface code Y basis”. Quantum8, 1310 (2024)

    2024

  6. [6]

    A denser planar surface code

    Guang Hao Low, William J. Huggins, Dominic W. Berry, Tanuj Khattar, Alec F. White, Nicholas C. Rubin, and Ryan Babbush. “A denser planar surface code” (2026). arXiv:2605.30455

    Pith/arXiv arXiv 2026

  7. [7]

    Surfacecodecompilation via edge-disjoint paths

    MichaelBeverland, VadymKliuchnikov, andEddieSchoute. “Surfacecodecompilation via edge-disjoint paths”. PRX Quantum3, 020342 (2022)

    2022

  8. [8]

    Realistic cost to execute practical quantum circuits using direct Clifford+T lattice surgery compilation

    Tyler LeBlond, Christopher Dean, George Watkins, and Ryan S. Bennink. “Realistic cost to execute practical quantum circuits using direct Clifford+T lattice surgery compilation”. ACM Transactions on Quantum Computing5, 1–28 (2024)

    2024

  9. [9]

    A game of surface codes: Large-scale quantum computing with lattice surgery

    Daniel Litinski. “A game of surface codes: Large-scale quantum computing with lattice surgery”. Quantum3, 128 (2019)

    2019

  10. [10]

    Magic state distillation: Not as costly as you think

    Daniel Litinski. “Magic state distillation: Not as costly as you think”. Quantum3, 205 (2019)

    2019

  11. [11]

    Picturing quantum software: An intro- duction to the zx-calculus and quantum compilation

    Aleks Kissinger and John van de Wetering. “Picturing quantum software: An intro- duction to the zx-calculus and quantum compilation”. Preprint. (2024). url:https: //github.com/zxcalc/book

    2024

  12. [12]

    Efficient magic state factories with a catalyzed |CCZ⟩to2|T⟩transformation

    Craig Gidney and Austin G. Fowler. “Efficient magic state factories with a catalyzed |CCZ⟩to2|T⟩transformation”. Quantum3, 135 (2019)

    2019

  13. [13]

    Flexible layout of surface code computations using AutoCCZ states

    Craig Gidney and Austin G. Fowler. “Flexible layout of surface code computations using AutoCCZ states” (2019). arXiv:1905.08916

    Pith/arXiv arXiv 2019

  14. [14]

    A SAT scalpel for lattice surgery: Representation and synthesis of subroutines for surface-code fault- tolerant quantum computing

    Daniel Bochen Tan, Murphy Yuezhen Niu, and Craig Gidney. “A SAT scalpel for lattice surgery: Representation and synthesis of subroutines for surface-code fault- tolerant quantum computing”. In 2024 ACM/IEEE 51st Annual International Sym- posium on Computer Architecture (ISCA). Pages 325–339. IEEE (2024)

    2024

  15. [15]

    TopoLS: Lattice surgery compilation via topological program transformations

    Junyu Zhou, Yuhao Liu, Ethan Decker, Justin Kalloor, Mathias Weiden, Kean Chen, Costin Iancu, and Gushu Li. “TopoLS: Lattice surgery compilation via topological program transformations” (2026). arXiv:2601.23109

    arXiv 2026

  16. [16]

    Topological quantum distillation

    H. Bombin and M. A. Martin-Delgado. “Topological quantum distillation”. Physical Review Letters97, 180501 (2006)

    2006

  17. [17]

    Scaling and logic in the colour code on a superconducting quantum processor

    Nathan Lacroix, Alexandre Bourassa, Francisco J. H. Heras, et al. “Scaling and logic in the colour code on a superconducting quantum processor”. Nature645, 614– 619 (2025)

    2025

  18. [18]

    Universal transversal gates with color codes: A simplified approach

    Aleksander Kubica and Michael E. Beverland. “Universal transversal gates with color codes: A simplified approach”. Physical Review A91, 032330 (2015)

    2015

  19. [19]

    Magic state cultivation: growing T states as cheap as CNOT gates

    Craig Gidney, Noah Shutty, and Cody Jones. “Magic state cultivation: growing T states as cheap as CNOT gates” (2024). arXiv:2409.17595

    Pith/arXiv arXiv 2024

  20. [20]

    Verylowoverheadfault-tolerantmagic state preparation using redundant ancilla encoding and flag qubits

    ChristopherChamberlandandKyungjooNoh. “Verylowoverheadfault-tolerantmagic state preparation using redundant ancilla encoding and flag qubits”. npj Quantum Information6, 91 (2020)

    2020

  21. [21]

    Low-overhead magic state distillation with color codes

    Seok-Hyung Lee, Felix Thomsen, Nicholas Fazio, Benjamin J. Brown, and Stephen D. Bartlett. “Low-overhead magic state distillation with color codes”. PRX Quantum6, 030317 (2025)

    2025

  22. [22]

    Quantum computing by color-code lattice surgery

    Andrew J. Landahl and Ciaran Ryan-Anderson. “Quantum computing by color-code lattice surgery” (2014). arXiv:1407.5103. 17

    Pith/arXiv arXiv 2014

  23. [23]

    Low-overhead quantum computing with the color code

    Felix Thomsen, Markus S. Kesselring, Stephen D. Bartlett, and Benjamin J. Brown. “Low-overhead quantum computing with the color code”. Physical Review Research 6, 043125 (2024)

    2024

  24. [24]

    New circuits and an open source decoder for the color code

    Craig Gidney and Cody Jones. “New circuits and an open source decoder for the color code” (2023). arXiv:2312.08813

    Pith/arXiv arXiv 2023

  25. [25]

    General tensor network decoding of 2D pauli codes

    Christopher T. Chubb. “General tensor network decoding of 2D pauli codes” (2021). arXiv:2101.04125

    Pith/arXiv arXiv 2021

  26. [26]

    Fast and accurate AI-based pre-decoders for color codes

    Jan Olle, Christopher Chamberland, Muyuan Li, and Igor Baratta. “Fast and accurate AI-based pre-decoders for color codes” (2026). arXiv:2607.10058

    Pith/arXiv arXiv 2026

  27. [27]

    Towards lattice surgery compilation for the color code using pipe diagrams

    Laura S. Herzog, Gilad Kishony, Robert Wille, and Austin Fowler. “Towards lattice surgery compilation for the color code using pipe diagrams” (2026). arXiv:2607.05501

    Pith/arXiv arXiv 2026

  28. [28]

    The boundaries and twist defects of the color code and their applications to topological quantum computation

    Markus S. Kesselring, Fernando Pastawski, Jens Eisert, and Benjamin J. Brown. “The boundaries and twist defects of the color code and their applications to topological quantum computation”. Quantum2, 101 (2018). arXiv:1806.02820

    Pith/arXiv arXiv 2018

  29. [29]

    Anyon condensation and the color code

    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”. PRX Quantum5, 010342 (2024)

    2024

  30. [30]

    Unfolding the color code

    Aleksander Kubica, Beni Yoshida, and Fernando Pastawski. “Unfolding the color code”. New Journal of Physics17, 083026 (2015). arXiv:1503.02065

    Pith/arXiv arXiv 2015

  31. [31]

    The ZX calculus is a language for surface code lattice surgery

    Niel de Beaudrap and Dominic Horsman. “The ZX calculus is a language for surface code lattice surgery”. Quantum4, 218 (2020). arXiv:1704.08670

    Pith/arXiv arXiv 2020

  32. [32]

    Flo- quetifying the colour code

    Alex Townsend-Teague, Julio Magdalena de la Fuente, and Markus Kesselring. “Flo- quetifying the colour code”. Electronic Proceedings in Theoretical Computer Science 384, 265–303 (2023). arXiv:2307.11136

    Pith/arXiv arXiv 2023

  33. [33]

    Unifying flavors of fault tolerance with the ZX calculus

    Hector Bombin, Daniel Litinski, Naomi Nickerson, Fernando Pastawski, and Sam Roberts. “Unifying flavors of fault tolerance with the ZX calculus”. Quantum8, 1379 (2024). arXiv:2303.08829

    Pith/arXiv arXiv 2024

  34. [34]

    XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime

    Julio C. Magdalena de la Fuente, Josias Old, Alex Townsend-Teague, Manuel Rispler, Jens Eisert, and Markus Müller. “XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime”. PRX Quantum6, 010360 (2025)

    2025

  35. [35]

    Triangular color codes on trivalent graphs with flag qubits

    Christopher Chamberland, Aleksander Kubica, Theodore J. Yoder, and Guanyu Zhu. “Triangular color codes on trivalent graphs with flag qubits”. New Journal of Physics 22, 023019 (2020)

    2020

  36. [36]

    High-threshold and low-overhead fault-tolerant quantum memory

    Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder. “High-threshold and low-overhead fault-tolerant quantum memory”. Nature627, 778–782 (2024)

    2024

  37. [37]

    Anyon theory and topological frustration of high-efficiency quantum low- density parity-check codes

    Keyang Chen, Yuanting Liu, Yiming Zhang, Zijian Liang, Yu-An Chen, Ke Liu, and Hao Song. “Anyon theory and topological frustration of high-efficiency quantum low- density parity-check codes”. Physical Review Letters135, 076603 (2025)

    2025

  38. [38]

    Generalized toric codes on twisted tori for quantum error correction

    Zijian Liang, Ke Liu, Hao Song, and Yu-An Chen. “Generalized toric codes on twisted tori for quantum error correction”. PRX Quantum6, 020357 (2025)

    2025

  39. [39]

    Tile codes: High-efficiency quantum codes on a lattice with boundary

    Vincent Steffan, Shin Ho Choe, Nikolas P. Breuckmann, Francisco Revson Fernandes Pereira, and Jens Niklas Eberhardt. “Tile codes: High-efficiency quantum codes on a lattice with boundary”. Physical Review Letters135, 170601 (2025). 18

    2025