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Useful magic states for gate teleportation must be Clifford-equivalent to diagonal states; the five-qubit distilled state is not among them.

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2026-07-10 06:15 UTC pith:XO7SJAEG

load-bearing objection Clean theory paper that pins down which magic states actually work for no-leakage gate teleportation and gives a usable synthesis algorithm plus a practical feedforward simplification.

arxiv 2607.08508 v1 pith:XO7SJAEG submitted 2026-07-09 quant-ph

Magic Gate Teleportation: Structure, Useful Resource States, and Simpler Feedforward

Yunzhe Zheng , Allen Zang , Aleksander Kubica This is my paper
classification quant-ph
keywords magic gate teleportationresource statesstabilizer codesClifford hierarchyfeedforward operatorsalgorithmic fault tolerancemagic-state distillation
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 non-Clifford gates, which are usually implemented by teleporting them from carefully prepared resource states. The paper studies magic gate teleportation (MGT): protocols that apply a non-Clifford gate to an arbitrary unknown input while revealing nothing about that input. After the final measurements are pushed backward, every such protocol is revealed to be an encoding of the input into a stabilizer code (chosen by the measurement outcomes) followed by a logical non-Clifford gate. From this structure the authors give an explicit construction that works for every resource state obtained by commuting Pauli rotations on a stabilizer state, together with an efficient circuit-synthesis algorithm. The same structure yields a converse: any resource state that can power an MGT protocol must itself be Clifford-equivalent to a diagonal state. In particular the single-qubit state produced by the classic five-qubit distillation protocol lies outside this class and cannot be used for MGT. The paper also characterises when the usual Clifford feed-forward can be replaced by a Pauli operator, which both simplifies certain distillation circuits and clarifies why algorithmic fault-tolerance schemes can treat some logical measurement outcomes as random coin flips.

Core claim

A pure state can serve as a resource for magic gate teleportation (a protocol that realises a non-Clifford gate on an arbitrary input without leaking information about that input) only if it is Clifford-equivalent to a diagonal state. For single-qubit resources this means the useful states lie on three great circles of the Bloch sphere; the five-qubit distilled state |F angle is therefore useless for MGT. The same characterisation holds for multi-qubit resources whenever the resource-side factors of the back-propagated measurements commute.

What carries the argument

The back-propagated view of an MGT protocol: after the Pauli-Z measurements are pushed to the beginning, the protocol encodes the input into a stabilizer code heralded by the outcomes and then applies a logical non-Clifford gate. This identity both supplies the constructive protocols (Theorem 1) and forces the resource-state constraints (Theorems 2–3).

Load-bearing premise

The multi-qubit converse assumes that the resource-side factors of the back-propagated measurements all commute with one another; the paper notes this covers every known protocol but leaves open whether non-commuting factors could allow non-diagonal useful states.

What would settle it

Exhibit a multi-qubit pure state that is not Clifford-equivalent to any diagonal state, yet still supports an MGT protocol whose back-propagated measurements have non-commuting resource-side Pauli factors, and verify that the protocol implements a non-Clifford gate on every input without revealing information.

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

If this is right

  • Any resource state obtained by commuting Pauli rotations of a stabilizer state can be turned into an explicit MGT circuit by the given synthesis algorithm.
  • Feed-forward operators become ordinary Pauli operators whenever the input is stabilised by a matching set of anti-commuting Paulis, allowing them to be absorbed into the Pauli frame.
  • In algorithmic fault-tolerance schemes, logical measurement outcomes that are 50/50 random can safely be replaced by coin flips because the corresponding feed-forward reduces to a Pauli update.
  • The five-qubit distilled |F angle state must be further converted (e.g., by parity measurement of two copies) before it can be used for gate teleportation under the no-leakage requirement.

Where Pith is reading between the lines

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

  • Magic-state factories that currently output |F angle-type states may need an extra conversion stage if their consumers insist on information-preserving teleportation.
  • Allowing controlled leakage of partial information about the input could enlarge the set of usable resource states beyond the diagonal class, at the cost of intermediate classical processing.
  • The same back-propagation identity may organise other resource theories of non-Clifford gates once the free operations are fixed to Clifford plus Z measurements.

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 / 5 minor

Summary. The paper develops a theory of magic gate teleportation (MGT): gate-teleportation protocols that implement non-Clifford unitaries on arbitrary input states without leaking information about those states, using free operations consisting of computational-basis preparation, Cliffords, and Pauli-Z measurements. After back-propagating the final measurements, MGT is shown (Lemma 1) to encode the input into a measurement-heralded stabilizer code and then apply a logical non-Clifford gate. Theorem 1 and Algorithm 1 construct explicit MGT protocols (and deterministic feed-forward) for any resource state obtained by commuting Pauli rotations applied to a stabilizer state. The converse direction (Theorems 2–3, via Lemma 2) proves that useful resource states must be Clifford-equivalent to diagonal states (single-qubit case exhaustive; multi-qubit case under commuting resource-side Paulis), so that the |F angle state from the [[5,1,3]] distillation protocol is not useful for MGT. Theorems 4 and Corollaries 2–3 give conditions under which feed-forward reduces to Pauli (or simpler unitary) operators, with applications to algorithmic fault tolerance and distillation circuits.

Significance. If the results hold, the paper supplies a clean structural characterization of a practically important class of gate-teleportation protocols and a sharp necessary condition on useful resource states. The constructive half (Theorem 1 + Algorithm 1) is immediately usable for circuit synthesis; the converse half cleanly separates “non-stabilizer” from “useful for information-preserving gate teleportation,” explaining why |F angle cannot be used directly. The feed-forward criteria give a transparent stabilizer-based account of why certain logical measurements in algorithmic fault tolerance can be replaced by coin tosses and absorbed into the Pauli frame, and they reduce the number of non-Pauli corrections needed in distillation factories. The derivations rely only on standard stabilizer and Clifford-hierarchy arguments, with no free parameters.

minor comments (5)
  1. Section IV.B / Theorem 3: the commuting-A_j hypothesis is already flagged by the authors as covering all known protocols; a short explicit remark that the unrestricted multi-qubit converse remains open would make the scope even clearer for non-specialist readers.
  2. Figure 1 caption and surrounding text: the three great circles on the Bloch sphere are described but not labelled with the corresponding Clifford conjugations of Z(θ)|+⟩; a one-line legend would improve readability.
  3. Appendix A, Algorithm 2: the encoding-circuit construction is standard but the notation for the standard-form blocks (A1,A2,B0,C,D,E) is introduced without a brief pointer to Gottesman’s original presentation; a single reference sentence would help.
  4. Examples 2–4 and Figure 2: the logical-operator choices after measurement are correct but the intermediate tableau updates are left implicit; a short parenthetical note that they follow the usual Aaronson–Gottesman update rule would remove any residual ambiguity.
  5. Typographical: occasional missing spaces around math operators (e.g., “P(θ)|s⟩”) and inconsistent use of “feedforward” vs “feed-forward”; pure presentation polish.

Circularity Check

0 steps flagged

No circularity: all theorems derive directly from the no-leakage condition, free-operation set, and standard stabilizer/Clifford formalism with no fitted parameters or load-bearing self-citations.

full rationale

The paper's derivation chain is self-contained. MGT is defined via free operations (Clifford + Z measurements) plus the recoverability condition (Eq. 5: Tr[M_j |η angle⟨η| ⊗ ρ_in] = 0 for arbitrary ρ_in). Lemma 1 follows immediately by commuting P(θ)⊗I through the back-propagated measurements. Theorem 1 constructs protocols by the explicit choice M = {P'_j ⊗ P'_j} and verifies the logical action by direct conjugation; Corollary 1 obtains the feed-forward by the same algebra. Theorem 2 (single-qubit) expands a general pure state as CX(φ)Z(θ)|+ angle and shows that φ otin (π/2)ℤ forces all single-qubit Pauli expectations nonzero, contradicting Eq. 5; the multi-qubit Theorem 3 invokes the same zero-expectation condition on a maximal commuting set {A_j} and applies the elementary equivalence of Lemma 2 (itself proved by Clifford diagonalization). Algorithm 1 is a standard tableau decoding of the resulting stabilizer code. Theorem 4 and Corollaries 2–3 are likewise direct commutation calculations under the stated stabilizer assumptions on the input. The only self-citations ([40], [53]) appear in the interpretive discussion of algorithmic fault tolerance (Sec. VI.B) and are not used to justify any theorem statement. No parameters are fitted, no uniqueness result is imported from prior author work, and no known empirical pattern is merely renamed. The multi-qubit restriction (commuting A_j) is explicitly flagged by the authors and does not create circularity inside the theorems as stated.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The paper is pure theory resting on the standard stabilizer formalism, the Clifford hierarchy, and the conventional free-operation set of fault-tolerant gate teleportation. No numerical free parameters are fitted. The only invented entity is the named class 'magic gate teleportation' itself, which is a definitional restriction of existing gate-teleportation protocols rather than a new physical object.

axioms (4)
  • standard math Stabilizer formalism and Gottesman-Knill theorem (Pauli stabilizers, tableau updates, Clifford conjugation of Paulis)
    Used throughout Sections II-V and Appendix A for encoding, decoding and measurement back-propagation.
  • standard math Definition of the Clifford hierarchy C_k (recursive conjugation of Paulis)
    Invoked for the hierarchy statement in Corollary 1 and for diagonal-gate characterizations.
  • domain assumption Free operations of MGT are computational-basis preparation, Clifford unitaries and single-qubit Z measurements
    Stated in Section II.B; defines the model in which 'useful' is evaluated.
  • domain assumption No information about the arbitrary input state may be revealed by the protocol (recoverability condition Tr[M_j |eta><eta| \otimes \rho_in]=0)
    Motivated by fault-tolerant computation and used as the key constraint in Theorems 2-3.
invented entities (1)
  • Magic gate teleportation (MGT) protocols no independent evidence
    purpose: Name the subclass of gate-teleportation protocols that implement non-Clifford gates on arbitrary inputs without leaking information about those inputs
    Definitional restriction of existing gate teleportation; no new physical resource is postulated, only a named class of protocols.

pith-pipeline@v1.1.0-grok45 · 21944 in / 2501 out tokens · 34488 ms · 2026-07-10T06:15:13.440215+00:00 · methodology

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read the original abstract

Quantum gate teleportation is a key technique in fault-tolerant quantum computation that uses resource states to implement logical gates. Here, we develop a theory of quantum gate teleportation protocols that implement non-Clifford gates on arbitrary input states without revealing any information about them; we refer to these protocols as magic gate teleportation (MGT). We uncover a hidden structure within MGT -- after backpropagating the Pauli measurements, MGT protocols can be viewed as encoding the input state into a stabilizer code heralded by the measurement outcomes, followed by a logical non-Clifford gate. Using this structure, we construct MGT protocols for any resource state obtained by applying commuting Pauli rotations to a stabilizer state, and provide an efficient algorithm for synthesizing their circuit implementations. Conversely, we prove that useful resource states for MGT, i.e., states that can be used for non-Clifford gates through MGT protocols, are necessarily Clifford-equivalent to diagonal states; in particular, the output state distilled from the $[\![5, 1, 3]\!]$ protocol is not useful for MGT. Finally, we identify conditions under which the feedforward operators can be implemented by Pauli operators, shedding light on the paradigm of algorithmic fault tolerance and simplifying the feedforward operations needed for quantum computing.

Figures

Figures reproduced from arXiv: 2607.08508 by Aleksander Kubica, Allen Zang, Yunzhe Zheng.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Magic gate teleportation (MGT) is a type of quantum gate teleportation that uses a non-stabilizer resource state [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Circuit implementation of deterministic MGT protocols obtained using Algorithm 1. Panels (a), (b) and (c) correspond to Exam [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) The 15-to-1 distillation circuit (adopted from Ref. [53]), with the stabilizer state [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages · 26 internal anchors

  1. [1]

    , Rn}, such that ⟨η|R j |η⟩= 0,∀R j ∈ R

    There exists a set ofnindependent commuting Pauli operatorsR={R 1, . . . , Rn}, such that ⟨η|R j |η⟩= 0,∀R j ∈ R

    work page

  2. [2]

    There exists a Clifford operatorCand a uni- tary diagonal in the computational basisD=P x∈{0,1}n eiθx |x⟩ ⟨x|, such that|η⟩=CD|+⟩ ⊗n

    work page

  3. [3]

    Proof.(1)→(2)

    There exists a set of non-trivial commuting Pauli oper- atorsPand a stabilizer state|s⟩that is not stabilized by any elements inP, such that|η⟩=P(θ)|s⟩. Proof.(1)→(2). SinceRis a maximal set of independent commuting Pauli operators, there is a Clifford unitaryCsuch thatC †RjC=Z j for allj. Let|ϕ⟩=C † |η⟩and write |ϕ⟩= X x∈{0,1}n ax |x⟩.(18) Then, the assu...

    work page

  4. [4]

    SIMPLE” We now discuss scenarios when the feedforward operators in MGT are “simple

    Using the equivalence between statements 1 and 2 in Lemma 2, we therefore have|η⟩=CD|+⟩ ⊗n for a Clifford operatorCand a diagonal unitaryD. V . CIRCUIT IMPLEMENTA TION OF MGT We now describe an explicit implementation of MGT. In principle, a deterministic MGT protocol can be implemented by applying a Clifford circuit to both the resource and input registe...

    work page

  5. [5]

    Shor, inProceedings of 37th Conference on Foundations of Computer Science, SFCS-96 (IEEE Comput

    P. Shor, inProceedings of 37th Conference on Foundations of Computer Science, SFCS-96 (IEEE Comput. Soc. Press, 1996) p. 56–65

    work page 1996

  6. [6]

    Preskill, Proceedings of the Royal Society of London

    J. Preskill, Proceedings of the Royal Society of London. Se- ries A: Mathematical, Physical and Engineering Sciences454, 385–410 (1998)

    work page 1998

  7. [7]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information(Cambridge university press, 2010)

    work page 2010

  8. [8]

    A. M. Dalzell, S. McArdle, M. Berta, P. Bienias, C.-F. Chen, A. Gily´en, C. T. Hann, M. J. Kastoryano, E. T. Khabiboulline, A. Kubica, G. Salton, S. Wang, and F. G. S. L. Brand ˜ao, Quantum Algorithms: A Survey of Applications and End-to-end Complexities(Cambridge University Press, 2025)

    work page 2025

  9. [9]

    P. W. Shor, Physical Review A52, R2493–R2496 (1995)

    work page 1995

  10. [10]

    A. M. Steane, Physical Review Letters77, 793–797 (1996)

    work page 1996

  11. [11]

    Gottesman, Physical Review A54, 1862–1868 (1996)

    D. Gottesman, Physical Review A54, 1862–1868 (1996)

    work page 1996

  12. [12]

    B. Zeng, A. Cross, and I. L. Chuang, Transversality versus Uni- versality for Additive Quantum Codes (2007), arXiv:0706.1382 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  13. [13]

    Eastin and E

    B. Eastin and E. Knill, Physical Review Letters102, 110502 (2009)

    work page 2009

  14. [14]

    Faist, S

    P. Faist, S. Nezami, V . V . Albert, G. Salton, F. Pastawski, 11 P. Hayden, and J. Preskill, Physical Review X10, 041018 (2020)

    work page 2020

  15. [15]

    Kubica and R

    A. Kubica and R. Demkowicz-Dobrza ´nski, Physical Review Letters126, 150503 (2021)

    work page 2021

  16. [16]

    Improved Simulation of Stabilizer Circuits

    S. Aaronson and D. Gottesman, Physical Review A70, 052328 (2004), arXiv:quant-ph/0406196

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  17. [17]

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Physical Review A86, 032324 (2012)

    work page 2012

  18. [18]

    E. T. Campbell, B. M. Terhal, and C. Vuillot, Nature549, 172–179 (2017)

    work page 2017

  19. [19]

    X. Zhou, D. W. Leung, and I. L. Chuang, Physical Review A 62, 052316 (2000), arXiv:quant-ph/0002039

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  20. [20]

    Gottesman and I

    D. Gottesman and I. L. Chuang, Nature402, 390 (1999)

    work page 1999

  21. [21]

    Bravyi and A

    S. Bravyi and A. Kitaev, Physical Review A71, 022316 (2005)

    work page 2005

  22. [22]

    A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery

    D. Litinski, Quantum3, 128 (2019), arXiv:1808.02892 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  23. [23]

    Surface code quantum computing by lattice surgery

    D. Horsman, A. G. Fowler, S. Devitt, and R. V . Meter, New Journal of Physics14, 123011 (2012), arXiv:1111.4022 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  24. [24]

    L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, Science Advances8, eabn1717 (2022)

    work page 2022

  25. [25]

    Z. He, A. Cowtan, D. J. Williamson, and T. J. Yoder, Extractors: QLDPC Architectures for Efficient Pauli-Based Computation (2025), arXiv:2503.10390 [quant-ph]

    work page arXiv 2025

  26. [26]

    Veitch, S

    V . Veitch, S. A. Hamed Mousavian, D. Gottesman, and J. Emer- son, New Journal of Physics16, 013009 (2014)

    work page 2014

  27. [27]

    Application of a resource theory for magic states to fault-tolerant quantum computing

    M. Howard and E. T. Campbell, Physical Review Letters118, 090501 (2017), arXiv:1609.07488 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  28. [28]

    Robustness of Magic and Symmetries of the Stabiliser Polytope

    M. Heinrich and D. Gross, Quantum3, 132 (2019), arXiv:1807.10296 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  29. [29]

    Lower bounds on the non-Clifford resources for quantum computations

    M. Beverland, E. Campbell, M. Howard, and V . Kliuch- nikov, Quantum Science and Technology5, 035009 (2020), arXiv:1904.01124 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  30. [30]

    X. Wang, M. M. Wilde, and Y . Su, Physical Review Letters124, 090505 (2020), arXiv:1812.10145 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  31. [31]

    Stabilizer R\'enyi entropy

    L. Leone, S. F. E. Oliviero, and A. Hamma, Physical Review Letters128, 050402 (2022), arXiv:2106.12587 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  32. [32]

    C. M. Dawson and M. A. Nielsen, Quantum Information and Computation6, 81 (2006), arXiv:quant-ph/0505030

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  33. [33]

    Kliuchnikov, D

    V . Kliuchnikov, D. Maslov, and M. Mosca, Physical Review Letters110, 190502 (2013)

    work page 2013

  34. [34]

    Efficient Clifford+T approximation of single-qubit operators

    P. Selinger, Efficient Clifford+T approximation of single-qubit operators (2014), arXiv:1212.6253 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  35. [35]

    Practical approximation of single-qubit unitaries by single-qubit quantum Clifford and T circuits

    V . Kliuchnikov, D. Maslov, and M. Mosca, IEEE Transactions on Computers65, 161 (2016), arXiv:1212.6964 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  36. [36]

    N. J. Ross and P. Selinger, Quantum Information and Computa- tion16, 901 (2016), arXiv:1403.2975 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  37. [37]

    A. J. Landahl and C. Cesare, Complex instruction set comput- ing architecture for performing accurate quantumZrotations with less magic (2013), arXiv:1302.3240 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  38. [38]

    Howard and H

    M. Howard and H. Dawkins, The European Physical Journal D 70, 55 (2016)

    work page 2016

  39. [39]

    H. Choi, F. T. Chong, D. Englund, and Y . Ding, Fault Tolerant Non-Clifford State Preparation for Arbitrary Rotations (2023), arXiv:2303.17380 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  40. [40]

    Akahoshi, K

    Y . Akahoshi, K. Maruyama, H. Oshima, S. Sato, and K. Fujii, PRX Quantum5, 010337 (2024)

    work page 2024

  41. [41]

    Theory and Architecture of Syndrome-Resolved Logical Gates

    N. Yoshioka, A. Seif, P. Groszkowski, A. Cross, and A. Javadi- Abhari, Theory and architecture of syndrome-resolved logical gates (2025), arXiv:2510.08290 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  42. [42]

    Transversal architecture for megaquop-scale quantum simulation with neutral atoms

    R. Ismail, I.-C. Chen, C. Zhao, R. Weiss, F. Liu, H. Zhou, S.- T. Wang, A. Sornborger, and M. Kornja ˇca, PRX Quantum7, 020343 (2026), arXiv:2509.18294 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  43. [43]

    Zheng and D

    Y . Zheng and D. E. Liu, Quantum9, 1858 (2025), arXiv:2412.04402 [quant-ph]

    work page arXiv 2025

  44. [44]

    H. Zhou, C. Zhao, M. Cain, D. Bluvstein, N. Maskara, C. Duck- ering, H.-Y . Hu, S.-T. Wang, A. Kubica, and M. D. Lukin, Na- ture646, 303 (2025), arXiv:2406.17653 [quant-ph]

    work page arXiv 2025

  45. [45]

    M. Cain, D. Bluvstein, C. Zhao, S. Gu, N. Maskara, M. Kali- nowski, A. A. Geim, A. Kubica, M. D. Lukin, and H. Zhou, Fast correlated decoding of transversal logical algorithms (2025), arXiv:2505.13587 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  46. [46]

    Serra-Peralta, M

    M. Serra-Peralta, M. H. Shaw, and B. M. Terhal, PRX Quantum 7, 010335 (2026)

    work page 2026

  47. [47]

    Stabilizer Codes and Quantum Error Correction

    D. Gottesman,Stabilizer Codes and Quantum Error Correc- tion, Ph.D. thesis, California Institute of Technology (1997), arXiv:quant-ph/9705052

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  48. [48]

    S. S. Hegde, K. R. K. Rao, and T. S. Mahesh, Pauli de- composition over commuting subsets: Applications in gate synthesis, state preparation, and quantum simulations (2016), arXiv:1603.06867 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  49. [49]

    B. Zeng, X. Chen, and I. L. Chuang, Physical Review A77, 042313 (2008)

    work page 2008

  50. [50]

    J. T. Anderson and T. Jochym-O’Connor, Quantum Information and Computation16, 771 (2016)

    work page 2016

  51. [51]

    S. X. Cui, D. Gottesman, and A. Krishna, Physical Review A 95, 012329 (2017), arXiv:1608.06596 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  52. [52]

    J. T. Anderson, Quantum8, 1370 (2024), arXiv:2212.05398 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  53. [53]

    Jones, Physical Review A87, 022328 (2013)

    C. Jones, Physical Review A87, 022328 (2013)

    work page 2013

  54. [54]

    Eastin, Physical Review A87, 032321 (2013)

    B. Eastin, Physical Review A87, 032321 (2013)

    work page 2013

  55. [55]

    Noise Thresholds for Higher Dimensional Systems using the Discrete Wigner Function

    W. van Dam and M. Howard, Physical Review A83, 032310 (2011), arXiv:1011.2497 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  56. [56]

    Qudit versions of the qubit "pi-over-eight" gate

    M. Howard and J. Vala, Physical Review A86, 022316 (2012), arXiv:1206.1598 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  57. [57]

    M. E. Beverland, A. Kubica, and K. M. Svore, PRX Quantum 2, 020341 (2021)

    work page 2021

  58. [58]

    Kitaev, Annals of Physics303, 2–30 (2003)

    A. Kitaev, Annals of Physics303, 2–30 (2003)

    work page 2003

  59. [59]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Journal of Mathematical Physics43, 4452–4505 (2002)

    work page 2002

  60. [60]

    Bombin, Physical Review X5, 031043 (2015)

    H. Bombin, Physical Review X5, 031043 (2015)

    work page 2015

  61. [61]

    E. T. Campbell, Quantum Science and Technology4, 025006 (2019)

    work page 2019

  62. [62]

    Kubica and M

    A. Kubica and M. Vasmer, Nature Communications13, 6272 (2022)

    work page 2022

  63. [63]

    S. Gu, E. Tang, L. Caha, S. H. Choe, Z. He, and A. Kubica, Communications in Mathematical Physics405, 85 (2024)

    work page 2024

  64. [64]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, Nature Com- munications5, 4213 (2014)

    work page 2014

  65. [65]

    Kandala, A

    A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Nature549, 242 (2017)

    work page 2017

  66. [66]

    A Quantum Approximate Optimization Algorithm

    E. Farhi, J. Goldstone, and S. Gutmann, A quantum approxi- mate optimization algorithm (2014), arXiv:1411.4028 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014