Pith. sign in

REVIEW 5 major objections 5 minor 1 cited by

The paper constructs a six-qutrit quantum error-correcting code whose logical AND gate is transversal, and a distance-4 concatenated extension.

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 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 18:47 UTC pith:6FN365LT

load-bearing objection Plausible new qutrit code with transversal AND, but the paper skips the stabilizer-conjugation argument that would close its main gap. the 5 major comments →

arxiv 2603.04548 v2 pith:6FN365LT submitted 2026-03-04 quant-ph cs.ET

Transversal AND in Quantum Codes

Christine Li , Lia Yeh This is my paper
classification quant-ph cs.ET MSC 81P7081P68 PACS 03.67.Pp
keywords quantum error correctionqutritstransversal gatesAND gateClifford+Texact circuit synthesisZX-calculusmagic 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.

The paper aims to show that the AND gate, irreversible on two-level qubits, becomes a reversible quantum operation on three-level qutrits, and that this reversibility can be packaged into an error-correcting code whose logical AND gate is transversal. Its central result is a [[6,2,2]] qutrit stabilizer code—two logical qutrits in six physical qutrits, distance 2—whose logical AND on the qubit subspace is implemented by a layer of three T and three T-dagger phase gates applied one qutrit at a time. The construction works by reading a symmetric compute-phase-uncompute circuit as a quantum code: the inner CX-only part defines a CSS code whose stabilizers and logical operators are read off directly, and an added X-type stabilizer raises the distance while preserving the logical AND. Concatenating this code with an [[8,1,2]] qutrit CSS code with transversal T yields a [[48,2,4]] code with the same transversal AND. This reverses the usual code-design order of business: rather than discovering which gates a code happens to support transversally, one starts from a desired non-Clifford gate and builds the code around it.

Core claim

The paper's main result is a new qutrit stabilizer code [[6,2,2]] with a transversal implementation of the logical AND gate. The AND gate is built from a symmetric T-depth-one circuit: an encoder made from CX gates, a layer of T and T-dagger gates, then the inverse encoder; reinterpreting this compute-phase-uncompute circuit as a code makes the inner CX portion a CSS code whose Z- and X-type stabilizer generators and logical operators can be read off directly. Adding the X-type stabilizer X1X2X3 and conjugating the circuit accordingly preserves the logical AND and yields distance 2, which the authors verify by checking the minimum weight of all logical operators. They further show that conca

What carries the argument

The central mechanism is the symmetric T-depth-one circuit decomposition: any unitary written as E (a circuit of CX gates), a layer of T and T-dagger gates, then E-dagger, can be reinterpreted as an encoder–decoder pair of a CSS error-correcting code. The inner CX network defines the CSS code, and its connectivity determines the stabilizer generators and logical operators; a 'pushing through the encoder' lemma, extended here to qutrits, moves logical gates through the full encoder to find their physical implementations. For AND, the target unitary is a |0>-controlled Z gate written in phase-gadget form and wrapped in outer Clifford gates, giving the full encoder E_AND. To raise the distance

Load-bearing premise

The construction assumes that the stabilizers, logical operators, and distance of the full code can be determined from the CX-only part of the encoder circuit, ignoring the outer Clifford and T gates when reading off the code; if that shortcut is invalid, the [[6,2,2]] parameters and the transversal AND property are not established.

What would settle it

Run the full six-qutrit encoder circuit—including every outer Clifford gate—through a stabilizer tableau computation and enumerate all logical operators. If any weight-1 logical operator appears, or if the transversal T/T-dagger layer does not act as AND on the encoded |0>/|1> subspace, then the claimed distance-2 code and its transversal gate are refuted.

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

If this is right

  • The [[6,2,2]] qutrit code provides a concrete example of a stabilizer code with a transversal non-Clifford entangling logical gate, using only 3 T and 3 T-dagger gates for the AND.
  • By concatenation with an existing [[8,1,2]] qutrit CSS code, one obtains a [[48,2,4]] qutrit code with a transversal AND gate, showing the distance can be systematically increased without losing the logical gate.
  • Any unitary admitting a symmetric T-depth-one compute-phase-uncompute decomposition can, by the same read-off procedure, be turned into a CSS code with that unitary as its transversal logical gate.
  • The Qubit Subspace Codes and projection gadgets allow a logical qubit to be carried by physical qutrits, with projective checks that keep the encoded states inside the qubit subspace.
  • The distillation and injection protocols give a deterministic way to apply AND in any qutrit CSS code, with corrections that are Clifford except for one single-qutrit diagonal gate.

Where Pith is reading between the lines

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

  • This suggests a general recipe: any multi-qutrit diagonal gate with a phase-gadget decomposition could seed a family of codes with built-in transversal non-Clifford gates; the paper demonstrates it for AND but does not explore the full family.
  • A natural next check is whether the [[6,2,2]] code, despite distance 2, offers a practical building block for fault tolerance when used as the inner code in concatenation, or whether the [[48,2,4]] code is where the practical advantage appears.
  • Because the qutrit AND emulation costs fewer non-Clifford resources than the best qubit Toffoli decompositions, the results point toward a mixed strategy—emulate qubit subcircuits on qutrit hardware and protect them with qutrit codes—though the paper does not provide a full resource comparison.
  • One testable extension would be to search directly for a qutrit code with parameters better than [[6,2,2]]/[[48,2,4]] for transversal AND, perhaps [[n,2,3]] or [[n,2,4]] without concatenation; if found, concatenation overhead could be reduced.

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

5 major / 5 minor

Summary. The paper proposes a qutrit [[6,2,2]] stabilizer code with a transversal implementation of the AND gate on its two logical qutrits, together with a concatenated [[48,2,4]] code. The construction starts from a symmetric T-depth-one Clifford+T decomposition of AND, interprets the inner CX (CSS) portion of the encoding circuit as a CSS code, and then uses outer Clifford gates of the full encoder to obtain a non-CSS code whose transversal T/T† layer should implement logical AND. The paper also gives T-counts for qutrit emulations of AND and n-ary AND, introduces 'Qubit Subspace Codes,' and presents magic-state distillation/injection protocols for qutrit CSS codes.

Significance. If the central construction is correct, this is a valuable addition to the small family of qutrit codes with non-Clifford transversal gates: a [[6,2,2]] code with transversal AND is a genuinely novel object, and the concatenation to [[48,2,4]] is a natural next step. The circuit-emulation T-count results (AND with T-count 3, n-ary AND with 3n−3) are also useful and are derived without fitted parameters. The paper's method—reverse-engineering a code from a symmetric circuit decomposition—is interesting and could inform future constructions. However, the main code claim is not currently demonstrated with the necessary rigor: several load-bearing checks are asserted rather than shown, and there is an unresolved CSS/non-CSS tension in the exposition.

major comments (5)
  1. [§3.2, §4.3, Eqs. (20)–(21)] The paper asserts that it is 'sufficient to focus on just the CSS portion of the circuit' for stabilizer generators, logical operators, and distance. This is not adequately justified. If the full encoder is E_AND = U_out ∘ E_pf, then the code is U_out(C_pf), whose stabilizer group is U_out S_pf U_out†, not the group in Eq. (20). Distance is invariant under the physical unitary U_out, so the distance check can legitimately be done on the inner CSS code, but this must be stated explicitly. The logical operators in Eq. (21) must also be checked to commute with the full stabilizer group; 'pushing through the full encoder' is a slogan, not a proof. Please provide the explicit full stabilizer group of the [[6,2,2]] code (or a direct derivation of Eq. (21) from E_AND) and show that Eq. (21) operators are logical and have weight ≥ 2.
  2. [§4.3, distance verification] The sentence 'We verify that the code distance is 2 by checking the minimum weight of all possible logical operators' is not a verifiable proof. For the central claim, the weight-1 and weight-2 logical operator enumeration should be displayed, or at least a precise algorithmic check (e.g., a small table or code snippet) should be included. Without this, the [[6,2,2]] parameter claim is unsupported.
  3. [§4.1, Eq. (15)] The identity in Eq. (15) is justified only by 'one can check that'. This identity is load-bearing: it is used to show that the modified circuit in Eq. (16) still preserves the logical AND gate. Please provide the ZX derivation or a direct matrix verification. A reader cannot be expected to take this on faith in a construction whose main claim depends on it.
  4. [§4.4, concatenation] The text says 'as both are qutrit CSS codes' when concatenating the [[6,2,2]] code with the inner [[8,1,2]] code. This contradicts §3.2, which explicitly states that the code corresponding to the full AND circuit is not a CSS code. The distance-product theorem for concatenated codes does not require both codes to be CSS, so the claim is likely salvageable; but the exposition must be corrected to explain which property of the inner code (e.g., transversal T with logical X/Z implemented by physical X/Z) is actually needed.
  5. [§3.1, Lemma 1 and abstract] The AND gate is described as a 'two-qutrit Clifford+T unitary,' but the truth tables in Table 1 and the circuit diagrams in Eqs. (5)–(8) appear to use three qutrits (a, b, c) with the third wire as an ancilla/output. For the [[6,2,2]] code with k=2, it is essential to specify exactly how the logical AND acts on two logical qutrits (e.g., as a two-qutrit unitary with a garbage output, or as a three-qutrit gate with one logical wire fixed to |0⟩). This ambiguity affects the definition of the claimed transversal logical gate and should be resolved.
minor comments (5)
  1. [Throughout] The notation J6,2,2K in the abstract and text should be typeset as [[6,2,2]].
  2. [§4.3, Eq. (20)] The notation for exponents is confusing (e.g., 'X 2_1 X2X 2_4 X5'). Please define exponents unambiguously, e.g., X_1^2 X_2 X_4^2 X_5.
  3. [§4.3, CSS stabilizers] If Eq. (20) is intended as a valid CSS stabilizer set, the commutation of the X-type stabilizer with the Z-type stabilizers should be checked explicitly; the current layout makes it hard to verify this by eye.
  4. [§4.4] The phrase 'transversal AND gate implemented by 24 T and 24 T† gates' should be accompanied by an explicit statement of which physical operation is applied to the 48 physical qutrits (e.g., a transversal layer of T/T† on each inner block).
  5. [§5–§6] The Qubit Subspace Codes and magic-state protocols are presented graphically with little supporting text. Since these are secondary results, they should be clearly marked as sketches or expanded, as appropriate for the journal.

Circularity Check

0 steps flagged

No significant circularity: the [[6,2,2]] code is a constructive synthesis from a chosen AND circuit, and the technical self-citations are independent graphical/ZX results rather than assumptions of the new code.

full rationale

The derivation chain is constructive rather than predictive. The AND gate is the stated target (Section 4, Eq. 8), the encoder E_AND is built from a symmetric T-depth-one decomposition of that gate, and the stabilizers and logical operators are read off from the inner CSS encoder and pushed through the outer Clifford layer (Eqs. 20-21). No parameter is fitted to data, and no closely related quantity is predicted from a fitted subset. The distance-2 claim is stated as a finite check ('checking the minimum weight of all possible logical operators') and is not derived from the input; even if that check is omitted or incorrect, it is an unverified assertion, not circularity. The paper does rely on prior work co-authored by one of the authors ([42], [61], [66], [80]), but those are general results about ZX-calculus, phase gadgets, and CSS code transformation that do not presuppose the [[6,2,2]] code or its transversal AND gate; they therefore count as independent support under the stated rules. There is an internal tension about whether the [[6,2,2]] code is CSS (Section 3.2 says 'not a CSS code' while Section 4.4 says 'both are qutrit CSS codes'), but that is a consistency/correctness concern, not a circularity: it does not make the conclusion equivalent to the assumptions. Finally, the near-identical AND emulation in Lemma 1 is explicitly attributed to [75], so the paper does not rename a known result as new. No step satisfies the quoted-equation reduction standard required for a circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters were fitted to data, and no new physical entities are postulated. The construction rests on ZX-calculus soundness, the phase-free/CSS correspondence, an ad hoc sufficiency assumption about focusing on the CSS portion, the existence of a cited inner CSS code, and standard ring-theoretic facts.

axioms (5)
  • standard math Qutrit ZX-calculus rewrite rules are sound, i.e. derivable equalities preserve linear maps.
    Used throughout the diagrammatic derivations, e.g. Section 3.2 Eqs. (9)-(10) and the phase-gadget rewrites in Section 4.2.
  • domain assumption Phase-free ZX diagrams correspond to CSS codes, and encoder normal forms determine stabilizers and logical operators.
    This is the basis for reading the code off the inner CX circuit in Section 3.2, citing [48].
  • ad hoc to paper The inner CSS (CX-only) portion of the circuit is sufficient to determine the stabilizers, logical operators, and distance of the full non-CSS code.
    Explicitly assumed in Section 3.2 without proof; it is load-bearing for the [[6,2,2]] construction.
  • domain assumption There exists a [[8,1,2]] qutrit CSS code with transversal T gate whose logical X and Z are physical X and Z.
    Used in Section 4.4 for the concatenated [[48,2,4]] code; cited to [13].
  • standard math Single-qutrit Clifford+T operators have matrix entries in the ring Z[1/(1-zeta)] with zeta = e^{2πi/9}.
    Used in Appendix D.1 to prove the qubit Hadamard gate cannot be exactly emulated by qutrit Clifford+T.

pith-pipeline@v1.3.0-alltime-deepseek · 28164 in / 17593 out tokens · 161508 ms · 2026-08-02T18:47:51.993647+00:00 · methodology

0 comments
read the original abstract

The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to $n$ qubits with T-count $3n-3$. Our main result is the construction of a novel qutrit $\mathopen{[\![} 6,2,2 \mathclose{]\!]}$ quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decomposition$\unicode{x2014}$composed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverse$\unicode{x2014}$of a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a $\mathopen{[\![} 48,2,4 \mathclose{]\!]}$ code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.

Figures

Figures reproduced from arXiv: 2603.04548 by Christine Li, Lia Yeh.

Figure 1
Figure 1. Figure 1: Rewrite rules for the qutrit ZX-calculus, reprinted from Figure 1 in [ [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Figure for a T-depth one decomposition of the qubit CCZ gate, reprinted [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Comparing Qubit and Qudit Encodings for EV Charging and Trip Assignment Problems

    quant-ph 2026-05 unverdicted novelty 6.0

    Qudit encodings for EV trip assignments cut the Hilbert space dimension exponentially and match or exceed qubit-based QAOA performance on constrained uni- and bi-directional charging problems.

Reference graph

Works this paper leans on

81 extracted references · 14 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Anderson, B.E., Sosa-Martinez, H., Riofrío, C.A., Deutsch, I.H., Jessen, P.S.: Accurate and robust unitary transformations of a high-dimensional quantum system. Phys. Rev. Lett.114, 240401 (Jun 2015).https: //doi.org/10.1103/PhysRevLett.114.240401,https://link.aps.org/doi/10. 1103/PhysRevLett.114.240401

    work page doi:10.1103/physrevlett.114.240401 2015

  2. [2]

    New Journal of Physics14(6), 063006 (Jun 2012).https://doi

    Anwar, H., Campbell, E.T., Browne, D.E.: Qutrit magic state distilla- tion. New Journal of Physics14(6), 063006 (Jun 2012).https://doi. org/10.1088/1367-2630/14/6/063006,https://iopscience.iop.org/article/ 10.1088/1367-2630/14/6/063006

    work page doi:10.1088/1367-2630/14/6/063006 2012

  3. [3]

    New Journal of Physics16(9), 093021 (Sep 2014).https://doi.org/10.1088/ 1367-2630/16/9/093021,http://dx.doi.org/10.1088/1367-2630/16/9/093021

    Backens, M.: The zx-calculus is complete for stabilizer quantum mechanics. New Journal of Physics16(9), 093021 (Sep 2014).https://doi.org/10.1088/ 1367-2630/16/9/093021,http://dx.doi.org/10.1088/1367-2630/16/9/093021

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

  4. [4]

    Scientific Reports9(1), 13389 (Sep 2019).https://doi.org/10

    Bækkegaard, T., Kristensen, L.B., Loft, N.J.S., Andersen, C.K., Petrosyan, D., Zinner, N.T.: Realization of efficient quantum gates with a superconducting qubit- Transversal AND in Quantum Codes 17 qutrit circuit. Scientific Reports9(1), 13389 (Sep 2019).https://doi.org/10. 1038/s41598-019-49657-1,https://doi.org/10.1038/s41598-019-49657-1

    doi:10.1038/s41598-019-49657-1 2019

  5. [5]

    ACM Transactions on Quan- tum Computing1(1), 1–25 (2020)

    Baker, J.M., Duckering, C., Gokhale, P., Brown, N.C., Brown, K.R., Chong, F.T.: Improved quantum circuits via intermediate qutrits. ACM Transactions on Quan- tum Computing1(1), 1–25 (2020)

    2020

  6. [7]

    Physical Review A 96(1), 012306 (Jul 2017).https://doi.org/10.1103/PhysRevA.96.012306,http: //link.aps.org/doi/10.1103/PhysRevA.96.012306

    Bocharov, A., Roetteler, M., Svore, K.M.: Factoring with qutrits: Shor’s algo- rithm on ternary and metaplectic quantum architectures. Physical Review A 96(1), 012306 (Jul 2017).https://doi.org/10.1103/PhysRevA.96.012306,http: //link.aps.org/doi/10.1103/PhysRevA.96.012306

    work page doi:10.1103/physreva.96.012306 2017

  7. [8]

    Booth, R.I., Carette, T.: Complete zx-calculi for the stabiliser fragment in odd prime dimensions. vol. 241, pp. 24:1–24:15. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022).https://doi.org/10.4230/LIPICS.MFCS.2022.24,https: //drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.24

    doi:10.4230/lipics.mfcs.2022.24 2022

  8. [9]

    Nature641(8063), 612–618 (May 2025).https://doi.org/10.1038/ s41586-025-08899-y,https://www.nature.com/articles/s41586-025-08899-y

    Brock, B.L., Singh, S., Eickbusch, A., Sivak, V.V., Ding, A.Z., Frunzio, L., Girvin, S.M., Devoret, M.H.: Quantum error correction of qudits beyond break-even. Nature641(8063), 612–618 (May 2025).https://doi.org/10.1038/ s41586-025-08899-y,https://www.nature.com/articles/s41586-025-08899-y

    2025

  9. [10]

    Nature Communications13(1), 3994 (Jul 2022).https://doi.org/10.1038/s41467-022-31638-0,https://doi.org/ 10.1038/s41467-022-31638-0

    Brown, T., Doucet, E., Ristè, D., Ribeill, G., Cicak, K., Aumentado, J., Simmonds, R., Govia, L., Kamal, A., Ranzani, L.: Trade off-free entanglement stabilization in a superconducting qutrit-qubit system. Nature Communications13(1), 3994 (Jul 2022).https://doi.org/10.1038/s41467-022-31638-0,https://doi.org/ 10.1038/s41467-022-31638-0

    doi:10.1038/s41467-022-31638-0 2022

  10. [11]

    Campbell, E.T.: The smallest interesting colour code (Sep 2016),https:// earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/

    2016

  11. [13]

    Campbell, E.T., Anwar, H., Browne, D.E.: Magic-state distillation in all prime dimensions using quantum reed-muller codes. Phys. Rev. X2, 041021 (Dec 2012).https://doi.org/10.1103/PhysRevX.2.041021,https://link.aps.org/ doi/10.1103/PhysRevX.2.041021

    doi:10.1103/physrevx.2.041021 2012

  12. [14]

    1103/PhysRevA.95.022316

    Campbell, E.T., Howard, M.: Unified framework for magic state distillation and multiqubitgatesynthesiswithreducedresourcecost.PhysicalReviewA95(2)(Feb 2017).https://doi.org/10.1103/physreva.95.022316,http://dx.doi.org/10. 1103/PhysRevA.95.022316

    doi:10.1103/physreva.95.022316 2017

  13. [16]

    Quantum Science and Tech- nology9(3), 035003 (apr 2024).https://doi.org/10.1088/2058-9565/ad37d4, https://doi.org/10.1088/2058-9565/ad37d4 18 C

    Cao, S., Bakr, M., Campanaro, G., Fasciati, S.D., Wills, J., Lall, D., Shteynas, B., Chidambaram, V., Rungger, I., Leek, P.: Emulating two qubits with a four-level transmon qudit for variational quantum algorithms. Quantum Science and Tech- nology9(3), 035003 (apr 2024).https://doi.org/10.1088/2058-9565/ad37d4, https://doi.org/10.1088/2058-9565/ad37d4 18 ...

    doi:10.1088/2058-9565/ad37d4 2024

  14. [17]

    Nature Communications13(1), 1166 (Mar 2022).https://doi.org/10.1038/s41467-022-28767-x,https://doi.org/ 10.1038/s41467-022-28767-x

    Chi, Y., Huang, J., Zhang, Z., Mao, J., Zhou, Z., Chen, X., Zhai, C., Bao, J., Dai, T., Yuan, H., Zhang, M., Dai, D., Tang, B., Yang, Y., Li, Z., Ding, Y., Oxen- løwe, L.K., Thompson, M.G., O’Brien, J.L., Li, Y., Gong, Q., Wang, J.: A pro- grammable qudit-based quantum processor. Nature Communications13(1), 1166 (Mar 2022).https://doi.org/10.1038/s41467-0...

    doi:10.1038/s41467-022-28767-x 2022

  15. [18]

    Chu, J., He, X., Zhou, Y., Yuan, J., Zhang, L., Guo, Q., Hai, Y., Han, Z., Hu, C.K., Huang, W., Jia, H., Jiao, D., Li, S., Liu, Y., Ni, Z., Nie, L., Pan, X., Qiu, J., Wei, W., Nuerbolati, W., Yang, Z., Zhang, J., Zhang, Z., Zou, W., Chen, Y., Deng, X., Deng, X., Hu, L., Li, J., Liu, S., Lu, Y., Niu, J., Tan, D., Xu, Y., Yan, T., Zhong, Y., Yan, F., Sun, X...

  16. [19]

    New Journal of Physics13(4), 043016 (Apr 2011).https://doi.org/10.1088/1367-2630/13/4/043016,http://dx.doi.org/ 10.1088/1367-2630/13/4/043016

    Coecke, B., Duncan, R.: Interacting quantum observables: categorical al- gebra and diagrammatics. New Journal of Physics13(4), 043016 (Apr 2011).https://doi.org/10.1088/1367-2630/13/4/043016,http://dx.doi.org/ 10.1088/1367-2630/13/4/043016

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

  17. [20]

    In: Coecke, B., Leifer, M

    Cowtan, A., Dilkes, S., Duncan, R., Simmons, W., Sivarajah, S.: Phase Gad- get Synthesis for Shallow Circuits. In: Coecke, B., Leifer, M. (eds.) Proceedings 16th International Conference on Quantum Physics and Logic, Chapman Univer- sity, Orange, CA, USA., 10-14 June 2019. Electronic Proceedings in Theoretical Computer Science, vol. 318, pp. 213–228. Open...

    doi:10.4204/eptcs.318.13 2019

  18. [21]

    Journal of Mathematical Physics56(3), 032202 (03 2015).https://doi.org/10

    Cui, S.X., Wang, Z.: Universal quantum computation with metaplectic anyons. Journal of Mathematical Physics56(3), 032202 (03 2015).https://doi.org/10. 1063/1.4914941

    2015

  19. [22]

    Dawkins, H., Howard, M.: Qutrit magic state distillation tight in some direc- tions. Phys. Rev. Lett.115, 030501 (Jul 2015).https://doi.org/10.1103/ PhysRevLett.115.030501,https://link.aps.org/doi/10.1103/PhysRevLett. 115.030501

    doi:10.1103/physrevlett 2015

  20. [23]

    org/abs/2502.01433

    Dutta, S., Wang, S., Baksi, A., Chattopadhyay, A., Maitra, S.: On exact space- depth trade-offs in multi-controlled toffoli decomposition (2025),https://arxiv. org/abs/2502.01433

    Pith/arXiv arXiv 2025

  21. [24]

    Electronic Proceedings in Theoretical Computer Science266, 147–163 (Feb 2018).https://doi.org/10.4204/eptcs.266.10,http://dx.doi.org/10

    Garvie, L., Duncan, R.: Verifying the smallest interesting colour code with quan- tomatic. Electronic Proceedings in Theoretical Computer Science266, 147–163 (Feb 2018).https://doi.org/10.4204/eptcs.266.10,http://dx.doi.org/10. 4204/EPTCS.266.10

    work page doi:10.4204/eptcs.266.10 2018

  22. [25]

    Scientific Reports5(1), 14671 (Oct 2015).https://doi.org/10.1038/srep14671, https://doi.org/10.1038/srep14671

    Gedik, Z., Silva, I.A., Çakmak, B., Karpat, G., Vidoto, E.L.G., Soares-Pinto, D.O., deAzevedo, E.R., Fanchini, F.F.: Computational speed-up with a single qudit. Scientific Reports5(1), 14671 (Oct 2015).https://doi.org/10.1038/srep14671, https://doi.org/10.1038/srep14671

    work page doi:10.1038/srep14671 2015

  23. [26]

    Quantum2, 74 (Jun 2018).https://doi.org/10.22331/q-2018-06-18-74,http://dx.doi.org/10

    Gidney, C.: Halving the cost of quantum addition. Quantum2, 74 (Jun 2018).https://doi.org/10.22331/q-2018-06-18-74,http://dx.doi.org/10. 22331/q-2018-06-18-74

    doi:10.22331/q-2018-06-18-74 2018

  24. [27]

    Gidney, C., Jones, N.C.: A cccz gate performed with 6 t gates (2021),https: //arxiv.org/abs/2106.11513

    Pith/arXiv arXiv 2021

  25. [28]

    Physi- cal Review A87(3) (Mar 2013).https://doi.org/10.1103/physreva.87.032332, http://dx.doi.org/10.1103/PhysRevA.87.032332 Transversal AND in Quantum Codes 19

    Giles, B., Selinger, P.: Exact synthesis of multiqubit clifford+t circuits. Physi- cal Review A87(3) (Mar 2013).https://doi.org/10.1103/physreva.87.032332, http://dx.doi.org/10.1103/PhysRevA.87.032332 Transversal AND in Quantum Codes 19

    doi:10.1103/physreva.87.032332 2013

  26. [29]

    Annals of Physics406, 54–70 (Jul 2019).https://doi.org/10

    Glaudell, A.N., Ross, N.J., Taylor, J.M.: Canonical forms for single-qutrit clif- ford+t operators. Annals of Physics406, 54–70 (Jul 2019).https://doi.org/10. 1016/j.aop.2019.04.001

    2019

  27. [30]

    Glaudell, A.N., Ross, N.J., Wetering, J.v.d., Yeh, L.: Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits (Aug 2024),http://arxiv.org/abs/2405.08136

    Pith/arXiv arXiv 2024

  28. [31]

    Godfrin, C., Ferhat, A., Ballou, R., Klyatskaya, S., Ruben, M., Wernsdorfer, W., Balestro, F.: Operating quantum states in single magnetic molecules: Im- plementation of grover’s quantum algorithm. Phys. Rev. Lett.119, 187702 (Nov 2017).https://doi.org/10.1103/PhysRevLett.119.187702,https://link.aps. org/doi/10.1103/PhysRevLett.119.187702

    doi:10.1103/physrevlett.119.187702 2017

  29. [32]

    In: Proceedings of the 46th International Symposium on Computer Architecture

    Gokhale, P., Baker, J.M., Duckering, C., Brown, N.C., Brown, K.R., Chong, F.T.: Asymptotic improvements to quantum circuits via qutrits. In: Proceedings of the 46th International Symposium on Computer Architecture. pp. 554–566. ACM, Phoenix Arizona (Jun 2019).https://doi.org/10.1145/3307650.3322253, https://dl.acm.org/doi/10.1145/3307650.3322253

    arXiv 2019

  30. [33]

    Na- ture Communications13(1), 7481 (Dec 2022).https://doi.org/10.1038/ s41467-022-34851-z,https://www.nature.com/articles/s41467-022-34851-z

    Goss, N., Morvan, A., Marinelli, B., Mitchell, B.K., Nguyen, L.B., Naik, R.K., Chen, L., Jünger, C., Kreikebaum, J.M., Santiago, D.I., Wallman, J.J., Sid- diqi, I.: High-fidelity qutrit entangling gates for superconducting circuits. Na- ture Communications13(1), 7481 (Dec 2022).https://doi.org/10.1038/ s41467-022-34851-z,https://www.nature.com/articles/s4...

    2022

  31. [34]

    Quantum Info

    Gosset, D., Kliuchnikov, V., Mosca, M., Russo, V.: An algorithm for the t-count. Quantum Info. Comput.14(15–16), 1261–1276 (Nov 2014)

    2014

  32. [35]

    Gottesman, D.: Stabilizer codes and quantum error correction. Ph.D. thesis, California Institute of Technology (1997).https://doi.org/10.48550/arXiv. quant-ph/9705052

    Pith/arXiv arXiv doi:10.48550/arxiv.quant-ph/9705052 1997

  33. [36]

    Physical Review A57(1),127–137(Jan1998).https://doi.org/10.1103/physreva.57.127,http: //dx.doi.org/10.1103/PhysRevA.57.127

    Gottesman, D.: Theory of fault-tolerant quantum computation. Physical Review A57(1),127–137(Jan1998).https://doi.org/10.1103/physreva.57.127,http: //dx.doi.org/10.1103/PhysRevA.57.127

    doi:10.1103/physreva.57.127

  34. [37]

    Gottesman, D.: 9.1 Concatenated Codes, p. 141–144. Textbook manuscript preprint (2024),http://www.cs.umd.edu/~dgottesm/QECCbook-2024.pdf

    2024

  35. [38]

    Gunderman, L.G.: Beyond integral-domain stabilizer codes. Phys. Rev. A112, 062430 (Dec 2025).https://doi.org/10.1103/4y6f-xx6s,https://link.aps. org/doi/10.1103/4y6f-xx6s

    work page doi:10.1103/4y6f-xx6s 2025

  36. [39]

    In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

    Hadzihasanovic, A., Ng, K.F., Wang, Q.: Two complete axiomatisations of pure- state qubit quantum computing. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science. p. 502–511. LICS ’18, Association for Computing Machinery, New York, NY, USA (2018).https://doi.org/10.1145/ 3209108.3209128,https://doi.org/10.1145/3209108.3209128

    arXiv 2018

  37. [40]

    Nature Communications14(1), 2242 (Apr 2023).https://doi.org/10.1038/s41467-023-37375-2,https://doi.org/10

    Hrmo, P., Wilhelm, B., Gerster, L., van Mourik, M.W., Huber, M., Blatt, R., Schindler, P., Monz, T., Ringbauer, M.: Native qudit entanglement in a trapped ion quantum processor. Nature Communications14(1), 2242 (Apr 2023).https://doi.org/10.1038/s41467-023-37375-2,https://doi.org/10. 1038/s41467-023-37375-2

    doi:10.1038/s41467-023-37375-2 2023

  38. [41]

    Science Ad- vances4(7),eaat9304(2018).https://doi.org/10.1126/sciadv.aat9304,https: //www.science.org/doi/abs/10.1126/sciadv.aat9304

    Hu, X.M., Guo, Y., Liu, B.H., Huang, Y.F., Li, C.F., Guo, G.C.: Beating the channel capacity limit for superdense coding with entangled ququarts. Science Ad- vances4(7),eaat9304(2018).https://doi.org/10.1126/sciadv.aat9304,https: //www.science.org/doi/abs/10.1126/sciadv.aat9304

    doi:10.1126/sciadv.aat9304 2018

  39. [42]

    Electronic Proceedings in Theoretical Computer Science384, 1–19 (Aug 2023).https://doi.org/10.4204/eptcs.384

    Huang, J., Li, S.M., Yeh, L., Kissinger, A., Mosca, M., Vasmer, M.: Graphical css code transformation using zx calculus. Electronic Proceedings in Theoretical Computer Science384, 1–19 (Aug 2023).https://doi.org/10.4204/eptcs.384. 1,http://dx.doi.org/10.4204/EPTCS.384.1 20 C. Li and L. Yeh

    work page doi:10.4204/eptcs.384 2023

  40. [43]

    Jones, C.: Low-overhead constructions for the fault-tolerant toffoli gate. Phys. Rev. A87, 022328 (Feb 2013).https://doi.org/10.1103/PhysRevA.87.022328, https://link.aps.org/doi/10.1103/PhysRevA.87.022328

    doi:10.1103/physreva.87.022328 2013

  41. [44]

    Joshi, R., Louw, J.C., Meth, M., Osborne, J.J., Mato, K., Su, G.X., Ringbauer, M., Halimeh, J.C.: Probing hadron scattering in lattice gauge theories on qudit quantum computers (2025),https://arxiv.org/abs/2507.12614

    Pith/arXiv arXiv 2025

  42. [45]

    Kabir, A., Nguyen, S., Ghosh, S., Kiran, T., Kim, I.H., Huang, Y.: Sdim: A qudit stabilizer simulator (2025),https://arxiv.org/abs/2511.12777

    Pith/arXiv arXiv 2025

  43. [46]

    Quantum9, 1752 (May 2025).https://doi.org/10.22331/ q-2025-05-21-1752,http://dx.doi.org/10.22331/q-2025-05-21-1752

    Khattar, T., Gidney, C.: Rise of conditionally clean ancillae for efficient quantum circuit constructions. Quantum9, 1752 (May 2025).https://doi.org/10.22331/ q-2025-05-21-1752,http://dx.doi.org/10.22331/q-2025-05-21-1752

    doi:10.22331/q-2025-05-21-1752 2025

  44. [47]

    PRX Quantum6, 040325 (Nov 2025).https://doi.org/10.1103/ 1gjs-2rhx,https://link.aps.org/doi/10.1103/1gjs-2rhx

    Khesin, A.B., Lu, J.Z., Shor, P.W.: Universal graph representation of stabi- lizer codes. PRX Quantum6, 040325 (Nov 2025).https://doi.org/10.1103/ 1gjs-2rhx,https://link.aps.org/doi/10.1103/1gjs-2rhx

    work page doi:10.1103/1gjs-2rhx 2025

  45. [48]

    Kissinger, A.: Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code) (Apr 2022),http://arxiv.org/abs/2204.14038

    Pith/arXiv arXiv 2022

  46. [49]

    Preprint (2024),https://github

    Kissinger, A., van de Wetering, J.: Picturing Quantum Software: An Introduction to the ZX-Calculus and Quantum Compilation. Preprint (2024),https://github. com/zxcalc/book/

    2024

  47. [50]

    New Journal of Physics17(8), 083026 (2015)

    Kubica, A., Yoshida, B., Pastawski, F.: Unfolding the color code. New Journal of Physics17(8), 083026 (2015)

    2015

  48. [51]

    Li, C., Preskill, J., Xu, Q.: Transversal dimension jump for product qldpc codes (2025),https://arxiv.org/abs/2510.07269

    arXiv 2025

  49. [52]

    Electronic Proceedings in Theoret- ical Computer Science426, 23–78 (Aug 2025).https://doi.org/10.4204/eptcs

    Li, S.M., Mosca, M., Ross, N.J., van de Wetering, J., Zhao, Y.: A complete and natural rule set for multi-qutrit clifford circuits. Electronic Proceedings in Theoret- ical Computer Science426, 23–78 (Aug 2025).https://doi.org/10.4204/eptcs. 426.2,http://dx.doi.org/10.4204/EPTCS.426.2

    doi:10.4204/eptcs 2025

  50. [53]

    Lysaght, L., Goubault, T., Sinnott, P., Mansfield, S., Emeriau, P.E.: Quantum circuit compression using qubit logic on qudits (2024),https://arxiv.org/abs/ 2411.03878

    Pith/arXiv arXiv 2024

  51. [54]

    In: 2023 IEEE International Conference on Quan- tum Software (QSW)

    Mato, K., Hillmich, S., Wille, R.: Compression of qubit circuits: Mapping to mixed- dimensional quantum systems. In: 2023 IEEE International Conference on Quan- tum Software (QSW). pp. 155–161 (2023).https://doi.org/10.1109/QSW59989. 2023.00027

    arXiv 2023

  52. [55]

    Mato, K., Ringbauer, M., Burgholzer, L., Wille, R.: Mqt qudits: A software frame- work for mixed-dimensional quantum computing (2024),https://arxiv.org/abs/ 2410.02854

    Pith/arXiv arXiv 2024

  53. [56]

    Nature Physics21(4), 570–576 (Apr 2025).https://doi.org/10.1038/ s41567-025-02797-w,https://doi.org/10.1038/s41567-025-02797-w

    Meth, M., Zhang, J., Haase, J.F., Edmunds, C., Postler, L., Jena, A.J., Steiner, A., Dellantonio, L., Blatt, R., Zoller, P., Monz, T., Schindler, P., Muschik, C., Ring- bauer, M.: Simulating two-dimensional lattice gauge theories on a qudit quantum computer. Nature Physics21(4), 570–576 (Apr 2025).https://doi.org/10.1038/ s41567-025-02797-w,https://doi.or...

    doi:10.1038/s41567-025-02797-w 2025

  54. [57]

    Nature Physics 19(12), 1780–1786 (Dec 2023).https://doi.org/10.1038/s41567-023-02226-w, https://www.nature.com/articles/s41567-023-02226-w

    Miao, K.C., McEwen, M., Atalaya, J., Kafri, D., Pryadko, L.P., Bengtsson, A., Opremcak, A., Satzinger, K.J., Chen, Z., Klimov, P.V., Quintana, C., Acharya, R., Anderson, K., Ansmann, M., Arute, F., Arya, K., Asfaw, A., Bardin, J.C., Bourassa, A., Bovaird, J., Brill, L., Buckley, B.B., Buell, D.A., Burger, T., Bur- kett, B., Bushnell, N., Campero, J., Chia...

    2023

  55. [58]

    Nakanishi, K.M., Todo, S.: Systematic construction of multi-controlled pauli gate decompositions with optimalt-count (2024),https://arxiv.org/abs/2410.00910

    Pith/arXiv arXiv 2024

  56. [59]

    Petit, L., Russ, M., Eenink, G.H., Lawrie, W.I., Clarke, J.S., Vandersypen, L.M., Veldhorst, M.: Design and integration of single-qubit rotations and two-qubit gates in silicon above one kelvin (2022)

    2022

  57. [60]

    Nature592(7853), 209–213 (apr 2021).https://doi.org/10.1038/ s41586-021-03318-4

    Pino, J.M., Dreiling, J.M., Figgatt, C., Gaebler, J.P., Moses, S.A., Allman, M.S., Baldwin, C.H., Foss-Feig, M., Hayes, D., Mayer, K., Ryan-Anderson, C., Neyenhuis, B.: Demonstration of the trapped-ion quantum CCD computer ar- chitecture. Nature592(7853), 209–213 (apr 2021).https://doi.org/10.1038/ s41586-021-03318-4

    2021

  58. [61]

    Electronic Proceedings in Theoretical Computer Science384, 220–264 (Aug 2023).https://doi.org/10.4204/eptcs.384.13,http://dx.doi.org/10

    Poór, B., Booth, R.I., Carette, T., van de Wetering, J., Yeh, L.: The qupit sta- biliser zx-travaganza: Simplified axioms, normal forms and graph-theoretic sim- plification. Electronic Proceedings in Theoretical Computer Science384, 220–264 (Aug 2023).https://doi.org/10.4204/eptcs.384.13,http://dx.doi.org/10. 4204/EPTCS.384.13

    doi:10.4204/eptcs.384.13 2023

  59. [62]

    Prakash, S.: Magic state distillation with the ternary golay code. Proceed- ings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476(2241), 20200187 (2020).https://doi.org/10.1098/rspa.2020.0187,https: //royalsocietypublishing.org/doi/abs/10.1098/rspa.2020.0187

    arXiv 2020

  60. [63]

    Physical Review A98(3) (Sep 2018).https://doi.org/10.1103/physreva.98.032304,http://dx.doi.org/10

    Prakash, S., Jain, A., Kapur, B., Seth, S.: Normal form for single-qutrit clif- ford+t operators and synthesis of single-qutrit gates. Physical Review A98(3) (Sep 2018).https://doi.org/10.1103/physreva.98.032304,http://dx.doi.org/10. 1103/PhysRevA.98.032304

    work page doi:10.1103/physreva.98.032304 2018

  61. [64]

    Electronic Proceedings in Theoretical Computer Science172, 68–91 (dec 2014).https://doi.org/10.4204/eptcs.172.6

    Ranchin, A.: Depicting qudit quantum mechanics and mutually unbiased qudit theories. Electronic Proceedings in Theoretical Computer Science172, 68–91 (dec 2014).https://doi.org/10.4204/eptcs.172.6

    work page doi:10.4204/eptcs.172.6 2014

  62. [65]

    Nature Physics18(9), 1053–1057 (Sep 2022).https://doi.org/10.1038/s41567-022-01658-0,https: //www.nature.com/articles/s41567-022-01658-0

    Ringbauer, M., Meth, M., Postler, L., Stricker, R., Blatt, R., Schindler, P., Monz, T.: A universal qudit quantum processor with trapped ions. Nature Physics18(9), 1053–1057 (Sep 2022).https://doi.org/10.1038/s41567-022-01658-0,https: //www.nature.com/articles/s41567-022-01658-0

    doi:10.1038/s41567-022-01658-0 2022

  63. [66]

    Electronic Proceedings in Theoretical Computer Science384, 142–170 (Aug 2023).https://doi.org/10.4204/eptcs.384.9,http: //dx.doi.org/10.4204/EPTCS.384.9

    Roy, P., van de Wetering, J., Yeh, L.: The qudit zh-calculus: Generalised tof- foli+hadamard and universality. Electronic Proceedings in Theoretical Computer Science384, 142–170 (Aug 2023).https://doi.org/10.4204/eptcs.384.9,http: //dx.doi.org/10.4204/EPTCS.384.9

    doi:10.4204/eptcs.384.9 2023

  64. [67]

    Physical Review A87(4), 042302 (Apr 2013).https://doi.org/10.1103/PhysRevA.87.042302 22 C

    Selinger, P.: Quantum circuits of t-depth one. Physical Review A87(4), 042302 (Apr 2013).https://doi.org/10.1103/PhysRevA.87.042302 22 C. Li and L. Yeh

    doi:10.1103/physreva.87.042302 2013

  65. [68]

    Sheldon, S., Magesan, E., Chow, J.M., Gambetta, J.M.: Procedure for systemat- ically tuning up cross-talk in the cross-resonance gate. Phys. Rev. A93, 060302 (Jun 2016).https://doi.org/10.1103/PhysRevA.93.060302

    doi:10.1103/physreva.93.060302 2016

  66. [69]

    Quantum Info

    Shende, V.V., Markov, I.L.: On the cnot-cost of toffoli gates. Quantum Info. Com- put.9(5), 461–486 (May 2009)

    2009

  67. [70]

    de Silva, N.: Efficient quantum gate teleportation in higher dimensions. Pro- ceedings of the Royal Society A: Mathematical, Physical and Engineering Sci- ences477(2251) (Jul 2021).https://doi.org/10.1098/rspa.2020.0865,http: //dx.doi.org/10.1098/rspa.2020.0865

    arXiv 2021

  68. [71]

    Physical Review X12(4) (Nov 2022).https://doi

    Srivastav, V., Valencia, N.H., McCutcheon, W., Leedumrongwatthanakun, S., De- signolle, S., Uola, R., Brunner, N., Malik, M.: Quick quantum steering: Overcoming loss and noise with qudits. Physical Review X12(4) (Nov 2022).https://doi. org/10.1103/physrevx.12.041023,http://dx.doi.org/10.1103/PhysRevX.12. 041023

    doi:10.1103/physrevx.12.041023 2022

  69. [73]

    Physical Review Letters78(11), 2252–2255 (Mar 1997).https://doi.org/ 10.1103/physrevlett.78.2252,http://dx.doi.org/10.1103/PhysRevLett.78

    Steane, A.M.: Active stabilization, quantum computation, and quantum state syn- thesis. Physical Review Letters78(11), 2252–2255 (Mar 1997).https://doi.org/ 10.1103/physrevlett.78.2252,http://dx.doi.org/10.1103/PhysRevLett.78. 2252

    doi:10.1103/physrevlett.78.2252 1997

  70. [74]

    Townsend-Teague, A., Meichanetzidis, K.: Simplification Strategies for the Qutrit ZX-Calculus (Jun 2022).https://doi.org/10.48550/arXiv.2103.06914

    Pith/arXiv arXiv doi:10.48550/arxiv.2103.06914 2022

  71. [75]

    Vlasov, A.Y.: Some notes on quantum information theory and emerging computing technologies (2015),https://arxiv.org/abs/1102.2890

    Pith/arXiv arXiv 2015

  72. [76]

    Wang, Q.: Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics (Mar 2018).https://doi.org/10.48550/arXiv.1803.00696,http://arxiv.org/ abs/1803.00696, arXiv:1803.00696

    Pith/arXiv arXiv doi:10.48550/arxiv.1803.00696 2018

  73. [77]

    Electronic Proceedings in Theoretical Computer Science172, 92–101 (Dec 2014).https: //doi.org/10.4204/eptcs.172.7,http://dx.doi.org/10.4204/EPTCS.172.7

    Wang, Q., Bian, X.: Qutrit dichromatic calculus and its universality. Electronic Proceedings in Theoretical Computer Science172, 92–101 (Dec 2014).https: //doi.org/10.4204/eptcs.172.7,http://dx.doi.org/10.4204/EPTCS.172.7

    work page doi:10.4204/eptcs.172.7 2014

  74. [78]

    Quantum6, 687 (apr 2022).https://doi.org/10.22331/q-2022-04-13-687,https://doi.org/ 10.22331%2Fq-2022-04-13-687

    Weggemans, J.R., Urech, A., Rausch, A., Spreeuw, R., Boucherie, R., Schreck, F., Schoutens, K., Minář, J., Speelman, F.: Solving correlation clustering with QAOA and a rydberg qudit system: a full-stack approach. Quantum6, 687 (apr 2022).https://doi.org/10.22331/q-2022-04-13-687,https://doi.org/ 10.22331%2Fq-2022-04-13-687

    doi:10.22331/q-2022-04-13-687 2022

  75. [79]

    New Jour- nal of Physics23(4), 043015 (apr 2021).https://doi.org/10.1088/1367-2630/ abf1b3

    van de Wetering, J.: Constructing quantum circuits with global gates. New Jour- nal of Physics23(4), 043015 (apr 2021).https://doi.org/10.1088/1367-2630/ abf1b3

    doi:10.1088/1367-2630/ 2021

  76. [80]

    Wetering, J.v.d., Yeh, L.: Building Qutrit Diagonal Gates from Phase Gadgets (Nov 2023).https://doi.org/10.48550/arXiv.2204.13681

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2204.13681 2023

  77. [81]

    Yeh,L.:Transdimensionalquantumcomputation:thegraphicalnovel.Ph.D.thesis, University of Oxford (2026)

    2026

  78. [82]

    Yeh, L., Huang, J., Kissinger, A., Li, S.M., van de Wetering, J.: A three-way normal form for stabiliser codes across zx diagrams, circuits, and tableaus (2026), in preparation

    2026

  79. [83]

    Yeh, L., van de Wetering, J.: Constructing All Qutrit Controlled Clif- ford+T gates in Clifford+T, p. 28–50. Springer International Publishing (2022).https://doi.org/10.1007/978-3-031-09005-9_3,http://dx.doi.org/ 10.1007/978-3-031-09005-9_3 Transversal AND in Quantum Codes 23

    work page doi:10.1007/978-3-031-09005-9_3 2022

  80. [84]

    Yu, N., Duan, R., Ying, M.: Five two-qubit gates are necessary for implement- ing the toffoli gate. Phys. Rev. A88, 010304 (Jul 2013).https://doi.org/10. 1103/PhysRevA.88.010304,https://link.aps.org/doi/10.1103/PhysRevA.88. 010304

    work page doi:10.1103/physreva.88 2013

Showing first 80 references.