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REVIEW 4 major objections 4 minor 47 references

The paper claims that every Pauli-based quantum error-correcting code over prime-dimensional qudits can be foliated into a graph state whose X measurements give fault-tolerant measurement-based quantum computing.

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2026-08-02 03:42 UTC pith:54HGJLWS

load-bearing objection A genuine, non-trivial qudit extension of foliation with an honest but restrictive noise model; the abstract overclaims fault tolerance for photonic platforms. the 4 major comments →

arxiv 2607.13784 v1 pith:54HGJLWS submitted 2026-07-15 quant-ph

Foliated Quantum Error Correction for Qudits

G\"ozde \"Ust\"un , Simon J. Devitt , Jason Saied This is my paper
classification quant-ph
keywords quditfoliationmeasurement-based quantum computationgraph statesPauli codesquantum error correctionqudit toric codedynamical codes
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.

The paper establishes a general recipe: starting from any quantum error-correcting code whose checks are Pauli operators over a prime-dimensional qudit, build a layered graph state whose single-qudit measurements reproduce the code's syndrome extraction round by round. The construction covers CSS stabilizer codes, non-CSS stabilizer codes such as the [[5,1,3]] perfect code, and dynamical codes such as a new qudit generalization of the CSS honeycomb code. The point is that one can perform fault-tolerant measurement-based quantum computing in higher-dimensional systems—potentially useful in photonics, where qudit states are easy to prepare and losses dominate—without designing a new code for each hardware platform. If the construction is correct, error detection is built into the graph state: certain products of measurement outcomes, called detectors, are guaranteed to be trivial unless errors occurred. Simulations of the foliated qudit toric code show a threshold that increases with qudit dimension and is comparable to or better than circuit-based implementations.

Core claim

The paper's central claim is that any Pauli-based code over prime-dimensional qudits admits a foliation: a graph state Λ(d,N,T,H) whose nodes are data qudits arranged in chains (one per code qudit, with alternating CZ±1 edges) and ancilla qudits that entangle with the supports of the code's checks at each time step. Measuring X on every data qudit and the appropriate Pauli on each ancilla teleports the logical state forward in time while extracting syndrome information. The crucial identity is the detector D(c,t): a product of the same ancilla's measurements two time steps apart, together with powers of X on the intermediate data qudits, chosen so that all Z factors cancel via the alternatin

What carries the argument

The central object is the foliated qudit graph state Λ (Definition 1 for CSS codes, Definition 4 for general Pauli codes): one linear chain state per data qudit, with CZ edges of alternating sign (−1)^t between consecutive time layers to cancel the unwanted Fourier factors of qudit teleportation, plus ancilla nodes connected to data qudits with weights given by the code's check matrices. The load-bearing identity is the detector D(c,t) = X_(1,c,t) X^{-1}_{(1,c,t+2)} ∏_q (X^{(H(t))_{cq}}_{(0,q,t+1)})^{(-1)^t}, which is a stabilizer of the graph state that commutes with every measurement; multiplying elementary detectors yields detectors for subsystem and dynamical codes. For non-CSS codes, an

Load-bearing premise

The fault-tolerance guarantee holds only when the graph state is prepared perfectly and all later noise is Pauli-Z; errors during graph-state preparation, such as photon loss or fusion failure, are not covered.

What would settle it

Test the foliated qudit toric code with a noise model that adds preparation errors (e.g., depolarizing noise or photon loss during graph-state growth) while using the identical decoding graph; if the logical error rate does not drop below the physical error rate as code distance increases, the practical fault-tolerance claim fails.

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

If this is right

  • Every qudit stabilizer, subsystem, and dynamical Pauli code in prime dimension has a measurement-based implementation with the same syndrome structure as its circuit version, so error-correction results transfer directly.
  • Qudit MBQC can exploit higher-dimensional codes: the foliated toric-code threshold rises with d, from about 0.023 at d=3 to about 0.088 at d=7919 in the simulated noise model.
  • The construction covers non-CSS codes by adding ancilla–ancilla edges and adjusting ancilla measurement bases, so it is not limited to CSS codes.
  • The qudit CSS honeycomb code generalization gives a Floquet example whose detectors span five time steps, showing that dynamical codes can be foliated.
  • Because only 2–3 layers of the graph state need be present at once, the protocol is adaptable to photonic generation and fusion, and the paper frames qudit fusion-based quantum computation as the next step.

Where Pith is reading between the lines

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

  • The authors note the equivalence only under a noise model with perfect graph-state preparation and post-preparation Pauli-Z errors. A natural extension is to analyze preparation errors, loss, and fusion failures in qudit photonic settings; until then, 'fault-tolerant' should be read as conditional.
  • The detector construction is algebraic and likely carries over to non-prime dimensions or finite fields with adjusted Weyl operators, though the paper does not claim this.
  • The reported toric-code threshold improvement may partly reflect the decoder heuristic's minimum-weight spanning-tree tie-breaking rather than the foliation itself; a head-to-head decoder comparison would isolate the source.
  • The layer-by-layer resource view suggests a testable extension: generate graph states sequentially with deterministic emitters and verify that detector failure rates scale with layer size and code distance as the threshold curves predict.

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

4 major / 4 minor

Summary. The paper proposes a framework for 'foliating' arbitrary Pauli-based quantum error-correcting codes over prime-dimensional qudits, extending earlier qubit constructions by Brown and Roberts. In the CSS case (Definition 1) it constructs a weighted qudit graph state whose measurements implement the code checks, with detectors given by Eq. (2). The non-CSS case is treated in Definition 4/S-A with detectors of the form (S.1). The authors claim applicability to stabilizer, subsystem, and dynamical codes, and give three examples: the qudit toric code, the [[5,1,3]] perfect qudit code, and a qudit generalization of the CSS honeycomb code. Under a simplified noise model—perfect preparation of the graph state followed by independent Pauli-Z errors—Proposition 3 maps the protocol to a circuit-based phenomenological noise model. Numerical HDRG-decoder thresholds for the foliated toric code are reported in Table I and claimed to increase with qudit dimension.

Significance. If the central claims are fully established, the paper would be a useful and nontrivial generalization of foliated MBQC to qudits, covering non-CSS and dynamical codes in a unified graph-state picture. The detector equations are derived from graph-state stabilizers rather than fitted to data, which is a strength, as is the concrete treatment of the toric-code unit cell and the non-CSS perfect-code example. The public simulation data and use of the sdim stabilizer simulator are also positive features. However, the advertised 'fault-tolerant' MBQC result is only demonstrated under a very restricted noise model, and the generality for arbitrary non-CSS/dynamical codes is asserted rather than proved. These issues are load-bearing for the paper's main claims, though they appear fixable by adding proofs and by qualifying the fault-tolerance statement.

major comments (4)
  1. [Abstract; Conclusion; Appendix A] The claim that the construction gives 'fault-tolerant measurement-based quantum computing' is not supported as stated. Proposition 3 and the simulations assume the graph state Lambda is prepared perfectly and only Pauli-Z errors occur afterwards. This excludes photon loss, failed fusion, and errors during the CZ entangling operations that are the dominant noise mechanisms for the photonic platform invoked throughout. The conclusion itself acknowledges that a photonic implementation would require joining smaller pieces via destructive fusion and defers that to future work. Please either prove fault tolerance under a preparation/loss-inclusive noise model or explicitly restrict the claim to phenomenological Z-noise after perfect preparation.
  2. [Definition 4; S-A; Eq. (S.1)] The central claim that the framework applies to 'any Pauli-based code' is not fully proved for non-CSS and dynamical codes. Proposition 3 is stated and proved only for the CSS case of Definition 1. For non-CSS codes, Definition 4 gives a construction and Eq. (S.1) states a detector form, but no general theorem demonstrates that these are stabilizers of the graph state commuting with all the prescribed measurements for arbitrary check matrices r(k), nor how the dynamical-case detectors are constructed in general. The examples are not a substitute for a general proof. Add a rigorous argument, or state precisely which classes are actually proven.
  3. [Appendix C; S-D E] The qudit CSS honeycomb code is presented as a new generalization of [26], but the paper offers only a one-sentence commutativity check ('This change ensures...'). Since this is the only dynamical-code example supporting the framework's claimed scope, the authors should verify that the period-6 schedule defines a valid Floquet code: the instantaneous stabilizer group, the deterministic measurement of the face stabilizers on the claimed rounds, and the logical operator structure. Without this, the dynamical-code example is not fully grounded.
  4. [Appendix B; Table I] The threshold comparison is internally inconsistent. The abstract and conclusion say the foliated toric-code thresholds are 'comparable' to the non-foliated version, while Appendix B says they are 'significantly higher' and attributes the difference to a modified decoder. The comparison is also not controlled: it compares a rotated toric code to the unrotated surface-code results of [18] with a different HDRG implementation. Please either provide a matched comparison under identical decoder and code geometry, or temper the claim to avoid overstating the numerical validation.
minor comments (4)
  1. [Definition 4; S-A] In Definition 4, 'r(t)' appears where the measurement round index k is meant; also the data-qudit index runs 0 to N-1 in Eq. (2)/(S.1) but 1 to N in Definition 1. Please reconcile the notation.
  2. [Appendix B; S-D E] Typos: 'readers unfamilar' in Appendix B; 'necesssary' in S-D E. These should be corrected.
  3. [Definition 2; main text] Time-boundary detectors are left as 'a straightforward exercise for the reader.' Since the memory-experiment simulations and the claimed equivalence in Proposition 3 depend on boundary initialization/terminal measurement conditions, these should be specified explicitly for a self-contained treatment.
  4. [Fig. 3(c) and Fig. S.10] The figures omit t=2 nodes. The captions should state clearly whether those nodes are absent from the graph or merely not shown, to avoid ambiguity about the graph-state structure used for the detector.

Circularity Check

0 steps flagged

No significant circularity: the foliated graph state and detectors are derived from the code's check matrices, and the numerical thresholds are simulation outputs rather than fitted inputs.

full rationale

The derivation chain is self-contained. Definition 1 and Definition 4 construct the foliated graph state directly from the code's check matrices H(t)/H~(k), and the detector formulas (Eq. 2 and Eq. S.1) are derived as graph-state stabilizers that commute with the prescribed X (or XZ^{-pX pZ}) measurements; they are not fitted to simulation data. The thresholds in Table I are outputs of the HDRG decoding simulation, not parameters used in the construction, so no fitted input is renamed as a prediction. The cited prior work ([14,15,17,18,26,40]) is external and is used as background, decoder, or simulator; the only self-citations ([34],[39]) concern photonic resource generation and are not load-bearing for the foliation construction. The acknowledged restriction in Appendix A to perfectly prepared graph states with post-preparation Pauli-Z errors is a scope limitation on the fault-tolerance claim, not a circular step. Thus no reduction of a claimed result to its inputs is present.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The central graph-state construction introduces no fitted constants; the thresholds in Table I are simulation outputs. The derivation relies on standard qudit graph-state stabilizer identities and the teleportation identity, while the non-CSS construction assumes checks in a round commute. Numerical threshold claims further depend on the HDRG decoder implementation and a simplified noise model. No new physical entities are introduced.

axioms (7)
  • standard math Qudit graph states have stabilizers X_i * prod_j Z_j^{w_ij} (Eq. 1).
    Used in Definition 1/2 and the supplementary detector derivations; standard qudit graph-state formalism from [27,41].
  • standard math For prime d, the qudit stabilizer formalism is a direct analogue of the qubit case with arithmetic mod d.
    Assumed throughout; cited to [28].
  • standard math Teleportation identity (Lemma 5): (langle+| x I) CZ^{+/-1} gives F^{+/-1} up to Pauli factors; alternating edge weights cancel F^2 != I.
    Proved in the supplementary material; central to the time-step structure of the foliation.
  • domain assumption Checks measured in the same round commute, so p_X*q_Z = p_Z*q_X for the ancilla-ancilla edge in Definition 4.
    Needed for the non-CSS edge construction; true for stabilizer checks and round-wise commuting Floquet schedules, but not derived in general.
  • ad hoc to paper Graph state is prepared perfectly; only Pauli-Z errors after preparation are considered, with each nonzero power having probability p/(d-1).
    Appendix A; used for all threshold numerics. Preparation errors and photon loss are not modeled.
  • domain assumption The HDRG decoder with the authors' minimum-spanning-tree correction rule succeeds at the quoted thresholds.
    Appendix B; no proof of decoder success, and the authors note that their modified correction rule changes the threshold values.
  • ad hoc to paper The qudit CSS honeycomb code obtained by replacing Z x Z with Z x Z-dagger in [26] is a valid CSS Floquet code.
    Appendix C; asserted with a commutation check but no general proof.

pith-pipeline@v1.3.0-alltime-deepseek · 17835 in / 15150 out tokens · 153878 ms · 2026-08-02T03:42:55.988443+00:00 · methodology

0 comments
read the original abstract

We present a framework for foliating any Pauli-based quantum error-correcting code over prime-dimensional qudits. For any such code, we obtain a qudit graph state that can be measured to perform fault-tolerant measurement-based quantum computing. Such a paradigm is of interest in platforms such as photonics, where measurement-based protocols are natural and high-dimensional states are readily available. We discuss several examples for arbitrary prime dimension $d$, such as the qudit toric code (stabilizer, CSS), the $d$-dimensional perfect $[[5,1,3]]$ code (stabilizer, non-CSS), and a straightforward $d$-dimensional generalization of the CSS honeycomb code (dynamical, CSS). Under a simple error model, we numerically calculate thresholds for the foliated qudit toric code and demonstrate that they are comparable to the non-foliated version.

Figures

Figures reproduced from arXiv: 2607.13784 by G\"ozde \"Ust\"un, Jason Saied, Simon J. Devitt.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) Teleportation of a single qudit state using a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: A linear chain state [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Examples of foliated qudit codes. (a) shows the unit [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The toric hexagonal lattice used for the CSS honey [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The logical vs. physical error rate of the foliated toric [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

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Reference graph

Works this paper leans on

47 extracted references · 1 canonical work pages

  1. [1]

    Fort >0, we add an edge from(0, q, t−1)to(0, q, t), with weight(−1) t

    We have the set ofdata qudit nodesD={(0, q, t) : 1≤q≤N,0≤t≤T}. Fort >0, we add an edge from(0, q, t−1)to(0, q, t), with weight(−1) t

  2. [2]

    stretched

    For0≤t≤T, we have the set ofancilla qudit nodesA t ={(1, c, t) : 1≤c≤r(t)}, with eachc corresponding to a check at timet(a row ofH(t)). For each such node(1, c, t), and each data quditq in the support ofc, we add an edge to(0, q, t)with weight(−1) tH(t) cq. To perform (a memory experiment in) MBQC, one sim- ply measures theXobservable on every qudit, modu...

  3. [3]

    Raussendorf and H

    R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett.86, 5188 (2001)

    2001

  4. [4]

    Raussendorf, D

    R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster 8 states, Physical review A68, 022312 (2003)

    2003

  5. [5]

    Bartolucci, P

    S. Bartolucci, P. Birchall, H. Bomb ´ ın, H. Cable, C. Daw- son, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and C. Sparrow, Fusion-based quantum computation, Nature Communications14, 912 (2023)

    2023

  6. [6]

    Bombin, I

    H. Bombin, I. H. Kim, D. Litinski, N. Nickerson, M. Pant, F. Pastawski, S. Roberts, and T. Rudolph, Interleaving: Modular architectures for fault-tolerant photonic quan- tum computing, arXiv preprint arXiv:2103.08612 (2021)

    Pith/arXiv arXiv 2021

  7. [7]

    Bombin, C

    H. Bombin, C. Dawson, T. Farrelly, Y. Liu, N. Nicker- son, M. Pant, F. Pastawski, and S. Roberts, Fault-tolerant complexes (2023), arXiv:2308.07844 [quant-ph]

    Pith/arXiv arXiv 2023

  8. [8]

    Broadbent, J

    A. Broadbent, J. Fitzsimons, and E. Kashefi, Universal blind quantum computation, in2009 50th Annual IEEE Symposium on Foundations of Computer Science(IEEE,

  9. [9]

    Morimae and K

    T. Morimae and K. Fujii, Blind topological measurement- based quantum computation, Nature communications3, 1036 (2012)

    2012

  10. [10]

    Hayashi and T

    M. Hayashi and T. Morimae, Verifiable measurement-only blind quantum computing with stabilizer testing, Physical review letters115, 220502 (2015)

    2015

  11. [11]

    J. F. Fitzsimons and E. Kashefi, Unconditionally verifi- able blind quantum computation, Physical Review A96, 012303 (2017)

    2017

  12. [12]

    Raussendorf, S

    R. Raussendorf, S. Bravyi, and J. Harrington, Long-range quantum entanglement in noisy cluster states, Phys. Rev. A71, 062313 (2005)

    2005

  13. [13]

    Raussendorf, J

    R. Raussendorf, J. Harrington, and K. Goyal, A fault- tolerant one-way quantum computer, Annals of Physics 321, 2242 (2006)

    2006

  14. [14]

    Raussendorf and J

    R. Raussendorf and J. Harrington, Fault-tolerant quan- tum computation with high threshold in two dimensions, Phys. Rev. Lett.98, 190504 (2007)

    2007

  15. [15]

    Raussendorf, J

    R. Raussendorf, J. Harrington, and K. Goyal, Topologi- cal fault-tolerance in cluster state quantum computation, New Journal of Physics9, 199 (2007)

    2007

  16. [16]

    A. Bolt, G. Duclos-Cianci, D. Poulin, and T. Stace, Foli- ated quantum error-correcting codes, Physical review let- ters117, 070501 (2016)

    2016

  17. [17]

    B. J. Brown and S. Roberts, Universal fault-tolerant measurement-based quantum computation, Physical Re- view Research2, 033305 (2020)

    2020

  18. [18]

    Andriyanova, D

    I. Andriyanova, D. Maurice, and J.-P. Tillich, New con- structions of css codes obtained by moving to higher al- phabets, arXiv preprint arXiv:1202.3338 (2012)

    Pith/arXiv arXiv 2012

  19. [19]

    Anwar, B

    H. Anwar, B. J. Brown, E. T. Campbell, and D. E. Browne, Fast decoders for qudit topological codes, New Journal of Physics16, 063038 (2014)

    2014

  20. [20]

    F. H. Watson, H. Anwar, and D. E. Browne, Fast fault- tolerant decoder for qubit and qudit surface codes, Phys- ical Review A92, 032309 (2015)

    2015

  21. [21]

    B. L. Brock, S. Singh, A. Eickbusch, V. V. Sivak, A. Z. Ding, L. Frunzio, S. M. Girvin, and M. H. Devoret, Quan- tum error correction of qudits beyond break-even, Nature 641, 612–618 (2025)

    2025

  22. [22]

    Ringbauer, M

    M. Ringbauer, M. Meth, L. Postler, R. Stricker, R. Blatt, P. Schindler, and T. Monz, A universal qudit quantum processor with trapped ions, Nature Physics18, 1053 (2022)

    2022

  23. [23]

    P. J. Low, B. White, and C. Senko, Control and readout of a 13-level trapped ion qudit, npj Quantum Information 11, 85 (2025)

    2025

  24. [24]

    D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, Quantum computation based on¡i¿d¡/i¿-level cluster state, Physical Review A68, 10.1103/physreva.68.062303 (2003)

    work page doi:10.1103/physreva.68.062303 2003

  25. [25]

    R. I. Booth, A. Kissinger, D. Markham, C. Meignant, and S. Perdrix, Outcome determinism in measurement-based quantum computation with qudits, Journal of Physics A: Mathematical and Theoretical56, 115303 (2023)

    2023

  26. [26]

    Romanova and W

    A. Romanova and W. D¨ ur, Measurement-based quantum computing with qudit stabilizer states, Quantum Science and Technology11, 015054 (2026)

    2026

  27. [27]

    Chau, Five quantum register error correction code for higher spin systems, Physical Review A56, R1 (1997)

    H. Chau, Five quantum register error correction code for higher spin systems, Physical Review A56, R1 (1997)

    1997

  28. [28]

    Davydova, N

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

    2023

  29. [29]

    Helwig, Absolutely maximally entangled qudit graph states (2013), arXiv:1306.2879 [quant-ph]

    W. Helwig, Absolutely maximally entangled qudit graph states (2013), arXiv:1306.2879 [quant-ph]

    Pith/arXiv arXiv 2013

  30. [30]

    D. Gottesman, Fault-tolerant quantum computation with higher-dimensional systems, inNASA International Con- ference on Quantum Computing and Quantum Communi- cations(Springer, 1998) pp. 302–313

    1998

  31. [31]

    Poulin, Stabilizer formalism for operator quantum er- ror correction, Physical review letters95, 230504 (2005)

    D. Poulin, Stabilizer formalism for operator quantum er- ror correction, Physical review letters95, 230504 (2005)

    2005

  32. [32]

    E. X. Fu and D. Gottesman, Error correction in dynamical codes, Quantum9, 1886 (2025)

    2025

  33. [33]

    S. S. Bullock and G. K. Brennen, Qudit surface codes and gauge theory with finite cyclic groups, Journal of Physics A: Mathematical and Theoretical40, 3481–3505 (2007)

    2007

  34. [34]

    Gimeno-Segovia, T

    M. Gimeno-Segovia, T. Rudolph, and S. E. Economou, Deterministic generation of large-scale entangled photonic cluster state from interacting solid state emitters, Physical review letters123, 070501 (2019)

    2019

  35. [35]

    Raissi, E

    Z. Raissi, E. Barnes, and S. E. Economou, Deterministic generation of qudit photonic graph states from quantum emitters, PRX Quantum5, 020346 (2024)

    2024

  36. [36]

    ¨Ust¨ un and S

    G. ¨Ust¨ un and S. J. Devitt, Comparing schemes for cre- ating qudit graph states from 16- and 128-dimensional hilbert space using donors in silicon, Phys. Rev. Res.8, 013343 (2026)

    2026

  37. [37]

    Lee and H

    S.-H. Lee and H. Jeong, Graph-theoretical optimization of fusion-based graph state generation, Quantum7, 1212 (2023)

    2023

  38. [38]

    Pankovich, A

    B. Pankovich, A. Neville, A. Kan, S. Omkar, K. H. Wan, and K. Br´ adler, Flexible entangled-state generation in lin- ear optics, Physical Review A110, 032402 (2024)

    2024

  39. [39]

    M. C. L¨ obl, L. A. Pettersson, S. Paesani, and A. S. Sørensen, Transforming graph states via bell state mea- surements, Quantum9, 1795 (2025)

    2025

  40. [40]

    D. E. Browne and T. Rudolph, Resource-efficient lin- ear optical quantum computation, Phys. Rev. Lett.95, 9 010501 (2005)

    2005

  41. [41]

    ¨Ust¨ un, E

    G. ¨Ust¨ un, E. G. Rieffel, S. J. Devitt, and J. Saied, Fu- sion for high-dimensional linear-optical quantum comput- ing with improved success probability, Physical Review Applied24, 044024 (2025)

    2025

  42. [42]

    Kabir, S

    A. Kabir, S. Nguyen, S. Ghosh, J. Keppens, T. Kiran, I. H. Kim, Y. Huang, and B. Sor´ ee, Sdim: A qudit stabilizer simulator (2026), arXiv:2511.12777 [quant-ph]

    Pith/arXiv arXiv 2026

  43. [43]

    Bahramgiri and S

    M. Bahramgiri and S. Beigi, Graph states under the action of local clifford group in non-binary case, arXiv preprint quant-ph/0610267 (2006)

    Pith/arXiv arXiv 2006

  44. [44]

    X. Zhou, D. W. Leung, and I. L. Chuang, Methodology for quantum logic gate construction, Physical Review A 62, 052316 (2000)

    2000

  45. [45]

    layers” or “time steps,

    M. A. Nielsen, Cluster-state quantum computation, Re- ports on Mathematical Physics57, 147 (2006). SUPPLEMENT AR Y MA TERIALS Sec. S-A formalizes the MBQC construction for the non-CSS case. The remaining sections give less formal expositions of the main ideas and discuss examples. S-A. FOLIA TING NON-CSS CODES We now formally discuss the non-CSS version o...

    2006

  46. [46]

    Fort >0, we add an edge from (0, q, t−1)to(0, q, t), with weight(−1)t

    We have the set ofdata qudit nodesD={(0, q, t) : 1≤q≤N,0≤t≤2K}. Fort >0, we add an edge from (0, q, t−1)to(0, q, t), with weight(−1)t

  47. [47]

    unit cell

    For0≤k≤K, we have the set ofancilla qudit nodesA k ={(1, c, k) : 1≤c≤r(t)}, with eachccorresponding to a check in measurement roundk(a row of ˜H(k)). We add the following edges: (a) For each such node(1, c, k), and each data quditqin the support ofc, leta=H(k) c,q andb=H(k) c,q+N , so that the part of the check supported onqhas the formX bZ a. We add an e...