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REVIEW 4 major objections 5 minor 76 references

Qudit Noisy Stabilizer Formalism

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Pauli-diagonal noise on qudit stabilizer states can be updated analytically under Clifford operations and generalized Pauli measurements, with cost linear in the initial state and exponential only in the final state.

desk verdict A genuine generalization of the qubit NSF to prime-power qudit graph states with sound appendix derivations; the only real soft spot is the unproven (but standard) stabilizer-to-graph equivalence for even prime powers. read the letter →

arxiv 2505.03889 v2 pith:ASEFVG3O submitted 2025-05-06 quant-ph

Paul Aigner , Maria Flors Mor-Ruiz , Wolfgang Dür This is my paper · ORCID
classification quant-ph
keywords quditnoisystabilizerformalismgraphstatesPauli-diagonalnoiseWeylmeasurementsCliffordoperationsdepolarizinggeneralizedBellpair
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces the qudit noisy stabilizer formalism, a bookkeeping scheme for stabilizer states in prime-power dimensions that are acted on by generalized Pauli-diagonal noise and then processed by Clifford gates and generalized Pauli measurements. The central move is to track the noise channels separately from the pure stabilizer part, so the full density matrix never has to be written down. The cost grows linearly with the number of initial qudits and noise terms, and exponentially only with the number of qudits left at the end, which makes protocols that distill or teleport a few entangled qudits analytically tractable. The authors demonstrate this by computing the fidelity of a generalized Bell pair produced from a noisy linear cluster state under two noise sources.

What carries the argument

The central object is a set of update rules: for a manipulation operator $O$ and a $Z$-type noise term $N$, $O N |G\rangle = \tilde N O |G\rangle$, with $\tilde N$ determined graphically. The rules are assembled from three building blocks—local multiplication, local complementation, and the $Z$ measurement—and every Weyl measurement reduces to these because its projector is Clifford equivalent to the $Z$ projector. The graphical measurement rules for $Z$, $Y$-type, $X$, $X(m)$, and $W(n,m)$ provide the graph transformations that accompany the updates. Global phases are dropped because the channels are Pauli-diagonal.

What would settle it

Run the update rules on a small system, for example three qudits of dimension $d=4$ in a linear cluster with depolarizing noise, and compare the final Bell-pair fidelity against full density-matrix evolution for the same measurement sequence; any mismatch at nonzero noise would falsify the update rules. Separately, exhibiting a $d=4$ stabilizer state that is not convertible to a graph state by single-qudit Clifford gates would falsify the claimed extension to all stabilizer states.

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Extended reading notes

Core claim

For a graph state in dimension $d = p^m$, any generalized Pauli noise term can be rewritten, using the graph stabilizers, as a product of $Z$-type operators. Each such operator commutes with a Clifford gate or a Weyl measurement up to a determined update: the noise operator is replaced by another $Z$-type operator whose support is read off from the current graph. Iterating these updates after every manipulation gives the exact final noisy state, and because prime-power stabilizer states are local Clifford equivalent to graph states, the same rules describe noisy stabilizer states generally. The only exponential step is applying the final updated noise maps to the small noiseless final state, so the formalism is efficient whenever the protocol ends in few qudits.

Load-bearing premise

The full-generality version of the formalism rests on the premise that every stabilizer state in prime-power dimension can be converted to a graph state by single-qudit Clifford gates; if that fails for some even prime-power dimensions, only the graph-state version remains valid.

Editorial extensions

If this is right

  • Protocols that end in a small entangled target—Bell-pair generation, entanglement swapping, purification—can be analyzed exactly under Pauli noise without constructing the full density matrix.
  • The fidelity of the generated generalized Bell pair is obtained as a closed analytic function of the measurement-order weight vector, the local dimension $d$, and the depolarizing parameter $\lambda$, so parameter scans are immediate.
  • The order of Weyl measurements on neighbouring qudits changes the final noise pattern even though the noiseless measurements commute, so protocols must specify and optimize an order.
  • The same bookkeeping applies to all stabilizer states in prime-power dimensions, and to graph states in arbitrary finite dimensions under operations that preserve the graph-state form.
  • Depending on how the depolarizing parameter scales with dimension, the adapted final fidelity can be highest for $d=2$, for large $d$, or for an intermediate dimension, so high-dimensional encodings are not universally better.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural experimental test is to realize the same linear-cluster Bell-pair protocol on qudits of dimension 2, 4, and 8 with controlled depolarizing noise and check whether the predicted intermediate-dimension optimum appears.
  • The noise–state separation suggests a continuous-variable counterpart: with the same commutation structure carried over to symplectic phase space, the formalism could describe Gaussian noise on continuous-variable cluster states.
  • For repeater protocols, the weight vector becomes a design parameter: one can search over measurement orders for the one that concentrates noise on terms that do not affect the target fidelity, without rerunning the full protocol each time.
  • Because the method tracks the exact noisy state rather than sampling from it, it can certify noise thresholds for small entangled states in regimes where Monte Carlo stabilizer simulation would need many samples.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This paper introduces a qudit generalization of the noisy stabilizer formalism (NSFd), extending the authors' earlier qubit formalism to prime-power local dimension d = p^m. The central idea is to treat Pauli-diagonal noise channels and stabilizer/graph states independently, updating the noise channels under Clifford operations and generalized Pauli (Weyl) measurements via a set of graphical update rules. The paper derives explicit graphical rules for Weyl measurements on graph states, noise update rules for the building-block Clifford operations, and an extension to general stabilizer states. It also discusses efficiency, giving a linear-in-initial-qudits and exponential-in-final-qudits scaling claim, and provides an extension to arbitrary finite dimensions via a linearized stabilizer formalism. As an application, the authors analyze the generation of a generalized Bell pair from a noisy linear cluster state under two depolarizing noise sources, deriving an analytic fidelity formula and a dimension-dependent noise parameter q_d fixed by matching Choi-Jamiołkowski fidelities.

Significance. If correct, the formalism is a useful analytical tool for noisy qudit protocols whose final state is small, extending the established qubit noisy stabilizer formalism to prime-power qudits. The paper is strong in its explicit derivations: Appendices B and C give detailed proofs of the measurement and noise update rules, and Appendix E provides a complete analytic fidelity formula. The dimension-dependent noise parameter q_d is fixed by matching Choi fidelities of qubit channels rather than fitted to the final fidelity, which is a methodological strength. The main risk concerns the breadth of the claimed applicability: the extension to all stabilizer states in prime-power dimensions rests on a cited but unproven local-Clifford equivalence, and the efficiency statement in Sec. V is not fully precise.

major comments (4)
  1. [§VI.A, first paragraph; Appendix A] The extension of the formalism from graph states to all stabilizer states in prime-power dimensions rests on the statement that every such stabilizer state is local Clifford equivalent to a graph state. The manuscript cites Refs. [27,28,45,46] but neither states the theorem precisely nor proves it for even prime-power dimensions, where Appendix A introduces different phase conventions (e.g., the S gate and H_even) and a Galois-ring construction. Since this is the only argument supporting the abstract's claim of applicability to all stabilizer states in prime-power dimensions, the authors should state the exact theorem and either provide a proof or give a reference that explicitly covers all p^m, including p=2 and m>1. If no such reference covers the even case, the applicability claim must be restricted to graph states or to the class for which the equivalence is proven.
  2. [§VI.A, paragraph beginning 'Alternatively, one can directly work with stabilizer states'] The direct stabilizer method assumes that for every measurement vertex v there exist stabilizer operators S_i whose local parts {S_i^v} generate the full local Weyl group, 'unless vertex v is disconnected in the local Clifford equivalent graph.' This assertion is load-bearing for the claimed treatment of Weyl measurements on general stabilizer states, but it is not proved and the condition is not made precise. The authors should provide a proof or a precise statement of this stabilizer-generation property, including the even-prime-power case, so that the direct method is more than a sketch.
  3. [§V and abstract] The complexity claim that the formalism 'scales linearly with the number of qudits in the initial state' is not stated with enough precision to be checked. The text says the number of different update rules is at most d×n, but for n single-qudit Pauli channels one has d^2×n noise terms, and each term is generally a product of elementary Z operators that must be combined. Please state the complexity of the full procedure with an explicit formula, including the d^2 factor and the cost of composing elementary updates, and specify whether d is treated as a constant or as an asymptotic parameter. The current wording conflates the number of elementary update rules with the number of noise terms and leaves the advertised exponential-in-final-size scaling incompletely specified.
  4. [§VI.B and Appendix D] The claim that the formalism 'remains applicable' for arbitrary finite dimensions is qualified in Appendix D by the condition that the Clifford equivalences for projectors do not require multiplicative inverses of zero-divisors, but the manuscript does not say which dimensions or which Weyl measurements satisfy this condition. Please specify the exact class of operations for which the arbitrary-dimension update rules are proven, and give at least one explicit example of a composite dimension and measurement where the restriction bites. Without this, the arbitrary-dimension part of the abstract is stronger than what the appendix actually establishes.
minor comments (5)
  1. [Eq. (29)] The block-vector notation 0_p and e_j is not defined, and e_j collides with the standard unit vector e_v introduced in Sec. II.D. Please define the index range of j and the length of each block explicitly.
  2. [§III.D, paragraph after Eq. (17)] The sentence 'A′ is the adjacency matrix of τ_{w0}(q)(G)' uses an undefined symbol q; it should presumably be r as in Eq. (17).
  3. [Appendix E3] The sentence defining the weight-vector entries with the function H(x)=1 for x≤0 and H(x)=0 for x>0 is easy to misread. Rewrite the entries w^k_1 directly as m + [k≤s] using Iverson brackets, which is what Eq. (29) appears to encode.
  4. [Appendix E2, step 4] The definition of the set U_h via tuples (i^j_{k_1},...,i^j_{k_h}) is not clear. Please rewrite the index set using standard set notation, since the final fidelity formula depends on this function π_h.
  5. [Throughout] The use of the symbol '9' to denote subtraction makes several equations difficult to read; replacing it with the standard minus sign would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the qudit update rules and Bell-pair fidelity are derived from commutation relations and channel equivalence, not fitted to the predicted output.

full rationale

The core derivation is self-contained. The qudit update rules (Sec. IV.A and Appendix C) are obtained by explicit commutation calculations from the Clifford/Weyl commutation table (Table II), not imported from the qubit NSF [41] or fitted to any output. The Weyl measurement rules (Sec. III and Appendix B) are likewise derived from projector identities and graph-state stabilizer manipulations. In the application, q_d in Eq. (26) is fixed by equating Choi-Jamiołkowski fidelities of m qubit depolarizing channels and one 2^m-dimensional depolarizing channel, an independent channel-equivalence condition; the final Bell-pair fidelity is then computed analytically via the derived weight-vector update rules (Appendix E), so Figure 6 is not a self-fulfilling fit. The extension to stabilizer states (Sec. VI.A) relies on the cited local-Clifford equivalence between stabilizer and graph states [27,28,45,46]; that is an external mathematical input, not a self-citation or a fitted parameter, so any concern about even-prime-power validity is a correctness/presentation gap rather than circularity. Self-citations [41-44] are background or prior qubit instances and do not carry the qudit derivation.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The formalism itself introduces no fitted free parameters and no new physical entities. It relies on standard stabilizer and graph-state background, on the Pauli-diagonal noise restriction, and on the local Clifford equivalence between stabilizer and graph states. The application uses model parameters r and q2 that are scanned or set by Choi-fidelity matching, not fitted to the target result.

assumptions (3)
  • standard math Every stabilizer state in prime-power dimension is local Clifford equivalent to a graph state.
    Invoked in Sec. VI.A to extend graph-state update rules to stabilizer states; cited to Refs. [27,28,45,46] but not proven in the paper.
  • domain assumption Noise channels considered are Pauli-diagonal; only then do noise channels commute and permit independent updating.
    Central restriction stated in Secs. I, IV, and VI.C; non-Pauli-diagonal channels break the independent-treatment mechanism.
  • domain assumption For arbitrary finite composite dimensions, update rules remain valid only when Clifford equivalences do not require multiplicative inverses of zero-divisors.
    Explicit caveat in Appendix D limiting the arbitrary-dimension extension.

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Cite this review

Pith. "Pith review of Qudit Noisy Stabilizer Formalism." pith.science (2026). https://pith.science/paper/ASEFVG3O

@misc{pith2026250503889,
  author       = {Pith},
  title        = {Pith review of: Qudit Noisy Stabilizer Formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ASEFVG3O}},
  note         = {Machine review of arXiv:2505.03889}
}
read the original abstract

We introduce the qudit Noisy Stabilizer Formalism, a framework for efficiently describing the evolution of stabilizer states in prime-power dimensions subject to generalized Pauli-diagonal noise under Clifford operations and generalized Pauli measurements. For arbitrary dimensions, the formalism remains applicable, though restricted to a subset of stabilizer states and operations. The computational complexity scales linearly with the number of qudits in the initial state and exponentially with the number of qudits in the final state. This ensures that when noisy qudit stabilizer states evolve via generalized Pauli measurements and Clifford operations to generate multipartite entangled states of a few qudits, their description remains efficient. We demonstrate this by analyzing the generation of a generalized Bell pair from a noisy linear cluster state subject to two distinct noise sources acting on each of the qudits.

Figures

Figures reproduced from arXiv: 2505.03889 by the authors.

Figure 2
Figure 2. Illustration of local multiplication (above) and local [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. Weighted graph, where dots represent vertices, lines [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Illustration of Weyl measurement operations acting [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Graphical representation of the W(n, m) measurement update rule constructed by concatenating the building-block update rules, which are applied in the following order: first L(m/n), followed by M(n), and last, a Z measurement. The vertex where the operations are applie…
Figure 5
Figure 5. Figure 5: Depiction of qudit linear chain with local depolariz [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: illustrates the behavior of the adapted fidelity as a function of the local dimension for different values of the dimension-independent parameter governing the strength of the depolarizing channel, 1 9 r. When the dimension-independent depolarizing effect is strong, i.…
Figure 7
Figure 7. Figure 7: Adapted fidelity F 1/m of a generalized Bell pair obtained via the side-to-side measurement strategy on a noisy linear cluster of size N = 10 in terms of the local dimension d = 2m, for different dimension-independent depolarizing parameters 1 9 r. The dimension-depend…

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Reference graph

Works this paper leans on

76 extracted references · 59 canonical work pages

  1. [41]

    P. J. Nadkarni and S. S. Garani, Phys. Rev. A 103, 042420 (2021)

    work page 2021

  2. [23]

    P. Liu, R. Wang, J.-N. Zhang, Y. Zhang, X. Cai, H. Xu, Z. Li, J. Han, X. Li, G. Xue, W. Liu, L. You, Y. Jin, and H. Yu, Phys. Rev. X13, 021028 (2023)

    2023

  3. [1]

    Single-qudit Pauli noise The action of a single-qudit Pauli noise channelEv on a quditv is described by Evρ = X z,x∈Fd λz,xZv(z)Xv(x)ρ (Zv(z)Xv(x))†, (6) whereλz,x≥ 0,P z,xλz,x = 1, andρ is a density matrix. Note that this form can be enforced via the random appli- cation of unitary gates before and after the application of a noise channel, without changi...

    work page

  4. [2]

    If this noise channel acts on a graph state, then it can be rewritten again in terms of onlyZ-type operators, using Eq

    Multiqudit Pauli channel A multiqudit Pauli channel is defined by Eρ = X z,x∈Fn d λz,xX(x)Z(z)ρ (X(x)Z(z))†, (8) where λz,x≥ 0 andP z,xλz,x = 1. If this noise channel acts on a graph state, then it can be rewritten again in terms of onlyZ-type operators, using Eq. (7). G. Noisy stabilizer formalism for qubits The noisy stabilizer formalism for qubits was ...

    work page

  5. [3]

    Bravyi and A

    S. Bravyi and A. Kitaev, Phys. Rev. A71, 022316 (2005)

    2005

  6. [4]

    All these operations map graph states to graph states

    Clifford operators The Clifford group acting on graph states is generated by the local complementation, the local multiplication, and theCZ gate. All these operations map graph states to graph states. The local multiplication M(m91) and the Z(z) Weyl operators commute if they act on different vertices. How- ever, if they act on the same vertex, then the u...

    work page

  7. [5]

    III the measurement operations are all equivalent up to Clifford operations

    Weyl measurements As seen in Sec. III the measurement operations are all equivalent up to Clifford operations. Therefore, it suffices to have the update rule for one measurement operation, here the WeylZ measurement such that the update rule for the rest of the Weyl measurements can be derived by concatenating the update rules of the corresponding Cliffor...

    work page

  8. [6]

    For each channel, for each noise term, each Pauli Z contribution is updated using the above update rules for O1

    So one starts with computing the effect ofO1 on both the noise channels and the noiseless graph state, leading to ˜Ei and ρ′ = |G′⟩⟨G′|, respec- tively. For each channel, for each noise term, each Pauli Z contribution is updated using the above update rules for O1. The update of the graph state is efficiently tractable thanks to the general- ized Gottesma...

    work page

Show all 76 references
  1. [7]

    Then the same procedure is performed for the next manipulation operatorO2, where now one updates the noise channels ˜Ei and the noiseless graph state ρ′ =|G′⟩⟨G′|, leading to ˜˜Ei and ρ′′ =|G′′⟩⟨G′′|. ... l. This is performed for the rest of the manipulation operators to retri...

  2. [8]

    F. H. E. Watson, H. Anwar, and D. E. Browne, Phys. Rev. A92, 032309 (2015)

    2015

  3. [9]

    R. S. Andrist, J. R. Wootton, and H. G. Katzgraber, Phys. Rev. A91, 042331 (2015)

    2015

  4. [10]

    Bombin and M

    H. Bombin and M. A. Martin-Delgado, Phys. Rev. A72, 032313 (2005)

    2005

  5. [11]

    Bourennane, A

    M. Bourennane, A. Karlsson, and G. Björk, Phys. Rev. A 64, 012306 (2001)

    2001

  6. [12]

    Miller, T

    D. Miller, T. Holz, H. Kampermann, and D. Bruß, Phys. Rev. A98, 052316 (2018)

    2018

  7. [13]

    G. M. Nikolopoulos, K. S. Ranade, and G. Alber, Phys. Rev. A73, 032325 (2006)

    2006

  8. [14]

    M. Doda, M. Huber, G. Murta, M. Pivoluska, M. Plesch, and C. Vlachou, Phys. Rev. Appl.15, 034003 (2021)

    2021

  9. [15]

    Srivastav, N

    V. Srivastav, N. H. Valencia, W. McCutcheon, S. Lee- dumrongwatthanakun, S. Designolle, R. Uola, N. Brun- ner, and M. Malik, Phys. Rev. X12, 041023 (2022)

    2022

  10. [16]

    Miguel-Ramiro and W

    J. Miguel-Ramiro and W. Dür, Phys. Rev. A98, 042309 (2018)

    2018

  11. [17]

    Neeley, M

    M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, A. N. Cleland, M. R. Geller, and J. M. Martinis, Science 325, 722 (2009), https://www.science.org/doi/pdf/10.1126/science.1173440

    2009 doi

  12. [18]

    A. R. Shlyakhov, V. V. Zemlyanov, M. V. Suslov, A. V. Lebedev, G. S. Paraoanu, G. B. Lesovik, and G. Blatter, Phys. Rev. A97, 022115 (2018)

    2018

  13. [19]

    Chizzini, F

    M. Chizzini, F. Tacchino, A. Chiesa, I. Tavernelli, S. Car- retta, and P. Santini, Phys. Rev. A110, 062602 (2024)

    2024

  14. [20]

    Ringbauer, M

    M. Ringbauer, M. Meth, L. Postler, R. Stricker, R. Blatt, P. Schindler, and T. Monz, Nat. Phys.18, 1053 (2022)

    2022

  15. [21]

    P. Hrmo, B. Wilhelm, L. Gerster, M. W. van Mourik, M. Huber, R. Blatt, P. Schindler, T. Monz, and M. Ring- bauer, Nat. Commun.14, 2242 (2023)

    2023

  16. [22]

    P. J. Low, B. M. White, A. A. Cox, M. L. Day, and C. Senko, Phys. Rev. Res.2, 033128 (2020)

    2020

  17. [24]

    Gottesman,The Heisenberg Representation of Quan- tum Computers, Tech

    D. Gottesman,The Heisenberg Representation of Quan- tum Computers, Tech. Rep. LA-UR-98-2848; CONF- 980788- (Los Alamos National Laboratory, Los Alamos, NM, 1998)

    1998

  18. [25]

    Omanakuttan, A

    S. Omanakuttan, A. Mitra, E. J. Meier, M. J. Martin, and I. H. Deutsch, PRX Quantum4, 040333 (2023)

    2023

  19. [26]

    Lindon, A

    J. Lindon, A. Tashchilina, L. W. Cooke, and L. J. LeBlanc, Phys. Rev. Appl.19, 034089 (2023)

    2023

  20. [27]

    Brecht, D

    B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, Phys. Rev. X5, 041017 (2015)

    2015

  21. [28]

    Zhong, H

    T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, New J. Phys.17, 022002 (2015)

    2015

  22. [29]

    N. T. Islam, C. C. W. Lim, C. Cahall, J. Kim, and D. J. Gauthier, Sci. Adv. 3, e1701491 (2017), https://www.science.org/doi/pdf/10.1126/sciadv.1701491

    2017 doi

  23. [30]

    De Beaudrap, Quantum Info

    N. De Beaudrap, Quantum Info. Comput. 13, 73–115 (2013)

    2013

  24. [31]

    Mansour and Z

    M. Mansour and Z. Dahbi, Int. J. Theor. Phys.59, 3876 (2020)

    2020

  25. [32]

    Aaronson and D

    S. Aaronson and D. Gottesman, Phys. Rev. A70, 052328 (2004)

    2004

  26. [33]

    Van Den Nes, Quantum Info

    M. Van Den Nes, Quantum Info. Comput.10, 258–271 (2010)

    2010

  27. [34]

    Grassl, A

    M. Grassl, A. Klappenecker, and M. Rotteler, in Pro- ceedings IEEE International Symposium on Information Theory (IEEE, 2002) p. 45

    2002

  28. [35]

    Schlingemann, Quantum Info

    D. Schlingemann, Quantum Info. Comput. 2, 307–323 (2002)

    2002

  29. [36]

    Englbrecht, T

    M. Englbrecht, T. Kraft, and B. Kraus, Quantum6, 846 (2022)

    2022

  30. [37]

    A. Keet, B. Fortescue, D. Markham, and B. C. Sanders, Phys. Rev. A82, 062315 (2010)

    2010

  31. [38]

    D.-S. Wang, D. T. Stephen, and R. Raussendorf, Phys. Rev. A95, 032312 (2017)

    2017

  32. [39]

    Gauthier, Recurrence Based Purification of Qudit Graph States, Master’s thesis, Delft University of Tech- nology (2021)

    S. Gauthier, Recurrence Based Purification of Qudit Graph States, Master’s thesis, Delft University of Tech- nology (2021)

    2021

  33. [40]

    P. J. Nadkarni and S. S. Garani, IEEE Trans. Quantum Eng. 2, 1 (2021)

    2021

  34. [42]

    S. Y. Looi and R. B. Griffiths, Phys. Rev. A84, 052306 (2011)

    2011

  35. [43]

    R. R. Vandermolen and D. Wright, Phys. Rev. A105, 032450 (2022)

    2022

  36. [44]

    D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, Phys. Rev. A 68, 062303 (2003)

    2003

  37. [45]

    Heinrich,On stabiliser techniques and their applica- tion to simulation and certification of quantum devices, Ph.D

    M. Heinrich,On stabiliser techniques and their applica- tion to simulation and certification of quantum devices, Ph.D. thesis, Universität zu Köln (2021)

    2021

  38. [46]

    Huang and P

    Y. Huang and P. Love, Phys. Rev. A99, 052307 (2019)

    2019

  39. [47]

    Brandl, Efficient and Noise-aware Stabilizer Tableau Simulation of Qudit Clifford Circuits, Master’s thesis, Jo- hannes Kepler University Linz (2024)

    N. Brandl, Efficient and Noise-aware Stabilizer Tableau Simulation of Qudit Clifford Circuits, Master’s thesis, Jo- hannes Kepler University Linz (2024). 12

    2024

  40. [48]

    M. F. Mor-Ruiz and W. Dür, Phys. Rev. A107, 032424 (2023)

    2023

  41. [49]

    The Choi-Jamiołkowski isomorphism relates channels (completely positive trace-preserving maps) with states, the so-called Choi-Jamiołkowski states

    is a measure of channel fidelity, which can be ob- tainedfromtheChoi-Jamiołkowskiisomorphism[50], and is such a state-independent measure, which lower bounds the fidelity of an arbitrary input state, subject to this channel [48]. The Choi-Jamiołkowski isomorphism relates chann...

  42. [50]

    M. F. Mor-Ruiz and W. Dür, IEEE J. Sel. Areas Com- mun. 42, 1793 (2024)

    2024

  43. [51]

    M. F. Mor-Ruiz, J. Wallnöfer, and W. Dür, Quantum9, 1605 (2025)

    2025

  44. [52]

    Merging-based quantum repeater,

    M. F. Mor-Ruiz, J. Miguel-Ramiro, J. Wallnöfer, T. Coopmans, and W. Dür, “Merging-based quantum repeater,” (2025), arXiv:2502.04450 [quant-ph]

    2025 arXiv

  45. [53]

    Gross, J

    D. Gross, J. Math. Phys.47, 122107 (2006)

    2006

  46. [54]

    L. E. Danielsen, Adv. Math. Commun.3, 329 (2009)

    2009

  47. [55]

    W. Dür, M. Hein, J. I. Cirac, and H.-J. Briegel, Phys. Rev. A72, 052326 (2005)

    2005

  48. [56]

    Gilchrist, N

    A. Gilchrist, N. K. Langford, and M. A. Nielsen, Phys. Rev. A71, 062310 (2005)

    2005

  49. [57]

    Jamiołkowski, Rep

    A. Jamiołkowski, Rep. Math. Phys.3, 275 (1972)

    1972

  50. [58]

    Gheorghiu, Phys

    V. Gheorghiu, Phys. Lett. A378, 505 (2014)

    2014

  51. [59]

    S. L. Braunstein and P. van Loock, Rev. Mod. Phys.77, 513 (2005)

    2005

  52. [60]

    S. D. Bartlett and B. C. Sanders, Phys. Rev. Lett.89, 207903 (2002)

    2002

  53. [61]

    Zhang, G

    J. Zhang, G. He, and G. Zeng, Phys. Rev. A80, 052333 (2009)

    2009

  54. [62]

    Continuous variable noisy stabilizer for- malism,

    P. Aigneret al., “Continuous variable noisy stabilizer for- malism,” (unpublished). 13 Appendix A: Background for even prime-power dimension In Sec. II, we have introduced the necessary mathematical background for odd prime-power dimensions. Here we present the differences wit...

  55. [63]

    Generalized Pauli group The generalized PauliX operator is the same as for the odd case, given by Eq. (1). TheZ operator is defined as Z(z) := X y∈Fd χ4(2yz)|y⟩⟨y|, whereχ4(x) =itr(x) and tr(t) is the trace of the linear map on theZ4-module R4m which acts asx7→x·t. Similar to ...

  56. [64]

    (2) for the odd case,K⊂ F2n d with liftα defines Kα :={(v, 2α(v))|v∈K}, which in turn sets the Abelian subgroupW (Kα) :={ωtr(α(v))W (v)|v∈K}

    Stabilizer formalism Here an isotropic subspace, Eq. (2) for the odd case,K⊂ F2n d with liftα defines Kα :={(v, 2α(v))|v∈K}, which in turn sets the Abelian subgroupW (Kα) :={ωtr(α(v))W (v)|v∈K}. To describe the liftα, we introduce β(v,w) =γ(v +w)9γ(v)9γ(w) + 2(vzwz), γ (v) :=v...

  57. [65]

    Moreover, if the power of our prime-power dimensionm is odd, then theH operator is defined asHeven =eiπ/4Hodd

    Clifford operations The generators of the Clifford group are the same as the ones presented in Table I, except theS operator, which is defined as S := X x∈Fd χ4(x2)|x⟩⟨x|. Moreover, if the power of our prime-power dimensionm is odd, then theH operator is defined asHeven =eiπ/4...

  58. [66]

    Z measurement For the following derivations we note that the computational basis is given as|b⟩ =|(1, 0),b⟩. First, we derive a useful expression for the projectorP (Zv,b ) P (Zv,b ) = 1 d X y∈Fd ¯χ(yb)Zv(y) = 1 d X y,k∈Fd ¯χ(yb)χ(yk)|k⟩v⟨k| = 1 d X k∈Fd |k⟩v⟨k| X y∈Fd ¯χ(y(b9...

  59. [67]

    We first show the local Clifford equivalence claimed in Eq

    Y-type measurement For the following derivations we note that theW (1,m ) basis elements areR(9m)|b⟩ =|(1,m ),b⟩. We first show the local Clifford equivalence claimed in Eq. (12), using Eq. (5) and Eq. (9), Lv(9m)P (Zv,b )Lv(m) = 1 d X y∈Fd Lv(9m)¯χ(yb)Zv(y)Lv(m) c.r. = 1 d X ...

  60. [68]

    The proof of the local Clifford equivalence of theX projector and theW (1, 1) projector presented in Eq

    X measurement For the following derivations we note that theX(1) basis elements areS(91)R(91)|b⟩ =|(0, 1),b⟩. The proof of the local Clifford equivalence of theX projector and theW (1, 1) projector presented in Eq. (14), is as follows [using Eq. (5) and Eq. (9)], wherew0 is a ...

  61. [69]

    We begin with showing the Clifford equivalence given in Eq

    X(m) measurement For the following derivations we note that theX(m) basis elements areM(m)S(91)R(91)|b⟩ =|(0,m ),b⟩. We begin with showing the Clifford equivalence given in Eq. (16), using Eq. (9), Mv(m)P (Xv,b )Mv(m91) = 1 d X y∈Fd ¯χ(by)Mv(m)Xv(y)Mv(m91) c.r. = 1 d X y∈Fd ¯χ...

  62. [70]

    First, we show the Clifford equivalence from Eq

    Z(n) measurement For the following derivations we note that theZ(n) basis elements areM(n91)|b⟩ =|(n, 0),b⟩. First, we show the Clifford equivalence from Eq. (18), using Eq. (9), Mv(n91)P (Zv,b )Mv(n) = 1 d X y∈Fd ¯χ(by)Mv(n91)Zv(y)Mv(n) c.r. = 1 d X y∈Fd ¯χ(by)Zv(ny) =P (Zv(n...

  63. [71]

    We first show the Clifford equivalence claimed in Eq

    W (n,m ) measurement For the following derivations we note that theW (n,m ) basis elements areR(9m/n)M(n91)|b⟩ =|(n,m ),b⟩. We first show the Clifford equivalence claimed in Eq. (19), using Eq.(9), Lv(9m/n)P (Zv(n),b )Lv(m/n) = 1 d X y∈Fd ¯χ(bw)Lv(9m/n)Zv(ny)Lv(m/n) c.r. = 1 d...

  64. [72]

    Local multiplication The update rule forM(m91), given in Eq

    Building blocks a. Local multiplication The update rule forM(m91), given in Eq. (22), directly follows from the commutation relation presented in Table II. b. Local complementation The update rule for L(m), given in Eq. (23), can be readily obtained via recalling the definitio...

  65. [73]

    So we can write the projection of an Weyl operatorO on a graph state|G⟩ always asP (Ov,b )|G⟩ =U†P (Zv,b )U|G⟩, where U is said Clifford

    Weyl measurement operators In the following, we make use of the fact that each Weyl projector is Clifford equivalent to theZ projector. So we can write the projection of an Weyl operatorO on a graph state|G⟩ always asP (Ov,b )|G⟩ =U†P (Zv,b )U|G⟩, where U is said Clifford. App...

  66. [74]

    The procedure to obtain the generalized Bell pair from the linear cluster state is to measure qudits 2 toN9 1 in theW (1, 1) basis

    Updated final noise maps Here we show the form of the updated noise channels acting on the generalized Bell pair. The procedure to obtain the generalized Bell pair from the linear cluster state is to measure qudits 2 toN9 1 in theW (1, 1) basis. We encode the order of these me...

  67. [75]

    (27) and Eq

    Fidelity of the generalized Bell pair The fidelity of the final generalized Bell pair isF =⟨G′|eEN··· eE1ρ′|G′⟩, where eEi are the updated noise maps following the descriptions of Eq. (27) and Eq. (28), andρ′ =|G′⟩⟨G′| with|G′⟩ being the generalized Bell pair between qudits 1 ...

  68. [76]

    Given a general noise term from the depolarizing channel, Eq

    Side-to-side strategy Here we derive the weight vector for the side-to-side strategy. Given a general noise term from the depolarizing channel, Eq. (25),Nj =Zj(z)Z(9Ajx) and the side-to-sideLσ, i.e.,σ = (2, 3,...,N 9 1), depending on which qudit j the noise operator acts on we...

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Reviewed August 15, 2026 · model on record in the stance chip above.