Pith. sign in

REVIEW 4 major objections 4 minor 1 cited by

On Exact Space-Depth Trade-Offs in Multi-Controlled Toffoli Decomposition

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read The paper derives exact formulas for the Toffoli depth of multi-controlled Toffoli decompositions with clean ancillae, proves the conditionally-clean-ancilla family cannot reach the logarithmic floor, and shows the floor is tight for all…

desk verdict Correct lower bound and a useful survey, but the paper's central exact trade-off formula contradicts its own example, and the negative result is overclaimed. read the letter →

arxiv 2502.01433 v3 pith:K56Y5FRF submitted 2025-02-03 quant-ph

Suman Dutta , Siyi Wang , Anubhab Baksi , Anupam Chattopadhyay , Subhamoy Maitra This is my paper · ORCID
classification quant-ph
keywords multi-controlledToffolidepthcleanancillaqubitsconditionallyancillaeClifford+Tdecompositionspace-depthtrade-offT-depthlowerbound
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

This paper pins down exactly how many layers of two-control Toffoli gates are needed to implement an $n$-controlled Toffoli ($n$-MCT) as a function of the number of clean ancilla qubits, using the standard Clifford+T gate set. For a family of decompositions based on conditionally clean ancillae—qubits that become clean during the computation and are then reused—it gives closed-form depth formulas: with $m=m_1+m_2$ ancillae, the complete Toffoli depth is $2\lfloor\log_2 m_1\rfloor+4k$ or $2\lfloor\log_2 m_1\rfloor+4k+2$, depending on which of two adjacent ranges contains $n$. The paper also proves a limitation of this technique: no matter how many ancillae are supplied, the depth stays strictly above $\lceil\log_2 n\rceil$. In a more general setting it shows that every Clifford+Toffoli decomposition of an $n$-MCT has Toffoli depth at least $\lceil\log_2 n\rceil$, and that a complete binary tree reaches that bound with $n-2$ ancillae. If correct, these results replace asymptotic estimates with exact resource counts for arithmetic and oracle circuits that contain multi-controlled Toffoli gates.

What carries the argument

The carrying object is the row representation of the conditionally-clean-ancilla schedule: Step I and Step II are drawn as rows $a \to \lceil a/2\rceil \to \cdots \to 1$, where each arrow is one Toffoli layer that halves the number of qubits carrying the accumulated AND information. The first row has $2m_1$ leaves, the remaining rows start from leftover control qubits, and the total number of leaves across rows is $n$; the number of layers $k$ fixes the size of the smaller MCT in Step III. The proof assumes that this smaller MCT overlaps Steps II and IV so that its net contribution to the total depth is 3 or 5. For the lower bound, the mechanism is the fan-in argument: each Toffoli gate can merge the information of at most two control qubits into one target, so any schedule is a binary tree with $n$ leaves, whose height is at least $\lceil \log_2 n\rceil$; a complete binary tree of that height gives the matching circuit.

What would settle it

A single explicit Clifford+Toffoli circuit for a 9-controlled Toffoli with Toffoli depth 3 would refute Theorem 3; a compiled instance of Construction 1 with $n=32$, $m_1=3$, $m_2=2$ whose scheduled depth is below 14 would refute Theorem 1's overlap assumption.

Watch

Extended reading notes

Core claim

The paper claims that the exact Toffoli depth of the conditionally-clean-ancilla $n$-MCT construction with $m = m_1 + m_2$ clean ancillae, taking $m_2 = 2$ for $m \geq 4$, is piecewise constant in $n$: on the interval $\big[(m_1 + 2^{\lfloor\log_2 m_1\rfloor - 1})2^{k-1} + k - 1,\; m_1 2^k + k - 2\big]$ it is $2\lfloor\log_2 m_1\rfloor + 4k$, and on the adjacent interval $[m_1 2^k + k - 1,\; (m_1 + 2^{\lfloor\log_2 m_1\rfloor - 1})2^k + k - 1]$ it is $2\lfloor\log_2 m_1\rfloor + 4k + 2$, for $k \geq 2$. The paper further claims that this construction keeps the depth strictly above $\lceil \log_2 n\rceil$, and that no Clifford+Toffoli decomposition of an $n$-MCT, regardless of technique or ancilla count, can go below $\lceil \log_2 n\rceil$. That lower bound is achieved by a complete binary tree using $n - 2$ ancillae and $n - 1$ Toffoli gates. Since single Toffoli gates can be replaced by measurement-based gadgets of T-depth 1, the same $\lceil \log_2 n\rceil$ value becomes the attainable T-depth floor with $2n - 2$ ancillae and T-count $4n - 4$; using logical-AND gadgets instead lowers the ancilla count to $n - 2$ while raising the T-depth to $\lceil \log_2 n\rceil + 1$.

Load-bearing premise

The depth formulas assume that the paper's row-by-row halving picture, with the middle smaller gate overlapping the surrounding two steps by exactly three or five Toffoli layers, captures every conditionally-clean-ancilla schedule; if a schedule outside this picture exists, the claimed impossibility for this family could fail.

Editorial extensions

If this is right

  • For any $n$ and any choice of $m_1$ in the construction, the Toffoli depth is known exactly, so a compiler can choose the ancilla count that meets a depth budget instead of working from an order-of-magnitude estimate.
  • The conditionally-clean-ancilla family is strictly bounded away from the floor: it can approach but never equal $\lceil \log_2 n\rceil$, so reaching the floor requires switching to the complete-binary-tree construction with $n-2$ ancillae.
  • No Clifford+Toffoli decomposition of an $n$-MCT, whatever the ancilla count or scheduling, can have Toffoli depth below $\lceil \log_2 n\rceil$.
  • Because a single Toffoli can be replaced by a measurement-based gadget of T-depth 1, the same floor transfers to T-depth; using logical-AND gadgets instead costs one extra T-depth layer and reduces the ancilla count from $2n-2$ to $n-2$.
  • The lower bound carries to any circuit containing an $n$-MCT as a subcircuit, including arithmetic, oracle circuits, and algebraic normal form synthesis, where the paper notes degree-$n$ Boolean functions can be built at Toffoli depth $\lceil \log_2 n\rceil$ with enough ancillae.

Reading between the lines

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

  • Inference: Theorem 2's impossibility is only as broad as the row-representation family; a conditionally-clean-ancilla circuit whose Step III overlaps the surrounding steps by less than three or five effective Toffoli layers would fall outside the formula and could in principle reach the floor.
  • Inference: The fan-in counting argument is really a statement about two-input AND accumulation, so the same $\lceil \log_2 n\rceil$ floor should hold for any reversible logic family built from two-input AND-like primitives, not only Clifford+Toffoli circuits.
  • Inference: Taking the complete-binary-tree achievability and Theorem 1 together suggests a sharp Pareto frontier—at every ancilla count below $n-2$, some depth above the floor is unavoidable—but the exact frontier outside the conditionally-clean family is not determined here.
  • Inference: A direct next step is to minimize the Theorem 1 formula over valid $m_1$ for each $n$ and total ancilla budget $m$; plotting that minimum against the complete-tree point would show the crossover where binary-tree decomposition becomes cheaper than the conditionally-clean construction.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies exact trade-offs between the number of clean ancilla qubits and the Toffoli depth needed to decompose an n-controlled Toffoli (n-MCT) gate using 2-controlled Toffoli gates. It proposes Construction 1, which splits m ancilla qubits into m1 for the initial accumulation and m2=2 for the final small MCT, and claims an exact closed-form depth formula in Theorem 1. From that formula the authors derive a negative result (Theorem 2) saying conditionally-clean-ancilla techniques cannot reach the exact lower bound ceil(log2 n), and then prove a general lower bound ceil(log2 n) for any Clifford+Toffoli decomposition (Theorem 3), with achievability via a complete binary tree. The main contribution is the claimed exact formula in Section IV B and its tabulated version in Table II.

Significance. If correct, the paper would provide a useful exact space-depth trade-off for MCT decomposition, extending the recent conditionally-clean-ancilla constructions of Nie et al. and Khattar and Gidney, and it would give a clean lower-bound sanity check for all Clifford+Toffoli decompositions. The paper is constructive, works with explicit formulas rather than asymptotics, and does not fit parameters to data; its comparisons with prior published circuits are valuable. However, the central exact formula is internally inconsistent with the paper's own example, so the main trade-off claim is not currently reliable. The general lower bound in Theorem 3 is plausible and potentially valuable, but its proof is too informal as written.

major comments (4)
  1. [Section IV B, Theorem 1 and Example 1] Theorem 1 is contradicted by Example 1. For n=32, m1=3 and k=3, n lies in the second interval of Theorem 1, whose formula gives 2*floor(log2 3) + 4*3 + 2 = 16. Example 1, with exactly the same parameters, computes the depth as 1 (Step I) + 2*5 (Steps II and IV) + 3 (Step III) = 14. The proof of Theorem 1 adds an effective Step III depth of 5 in both intervals, while Lemma 5 says the effective depth is either 3 or 5 and Example 1 requires 3. No condition is given for choosing 3 versus 5. Since the theorem is advertised as exact, this is a load-bearing error: the formula, its proof, and the corresponding row of Table II are all affected.
  2. [Section IV B, Lemma 5] Lemma 5 first states that the Step III Toffoli depths are 4 log2(k + m1 - 1) and 4 log2(k + m1), then immediately says that the effective Toffoli depth of Step III is 'either 3 or 5'. These statements are mutually incompatible: the logarithmic expressions are not integers in general and are not used anywhere in the proof of Theorem 1. This matters because Theorem 1's additive constant comes exactly from the '5' option, and the paper never explains when the overlap yields 3 rather than 5. The lemma needs a correct and consistent derivation before Theorem 1 can be trusted.
  3. [Section IV C, Theorem 2] The statement of Theorem 2 claims that 'using the conditionally clean ancillae technique' the exact Toffoli depth must be strictly greater than ceil(log2 n), irrespective of the number of ancilla. The proof, however, analyzes only the depth formula derived for Construction 1 and its specific row representation. A different scheduling of the conditionally clean ancilla accumulation is not ruled out, so the negative result overgeneralizes. In addition, the proof inherits the disputed Step III constant from Theorem 1 and cites 'Proposition 1' where it presumably means Theorem 1 or Proposition 3. The claim should be restated as a property of Construction 1, or a genuinely general argument must be supplied.
  4. [Section V, Theorem 3] The proof of the general lower bound is only an informal binary-tree argument. To justify an exact lower bound for arbitrary Clifford+Toffoli decompositions, the paper needs a formal argument that every such circuit implementing an n-MCT must contain a qubit whose final value depends on all n control values, and that each Toffoli gate can at most double the size of that dependency set. The current text asserts without proof that 'the Toffoli depth of the n-MCT is also at least ceil(log2 n)' from the height of a binary tree with n leaves. The lower bound is likely correct, but the proof as written is not sufficient for the exactness claimed in Theorem 3 and Corollaries 1 and 2.
minor comments (4)
  1. [Abstract] There is a typo: 'hereform' should be 'hereafter', and the mathematical notation 'ceil(log2 n)' should be typeset consistently with the main text.
  2. [Section IV C, proof of Theorem 2] The phrase 'Let us first proof the item 1' should read 'Let us first prove item 1', and the reference to 'Proposition 1' should be corrected to the appropriate theorem or proposition.
  3. [Table II] The interval notation in the row labeled 'Ours' is difficult to parse; the paper would benefit from a displayed definition of the two cases with explicit quantifiers for k and a separate explanation of the notation (m1 + 2^{floor(log2 m1)-1})2^{k-1}.
  4. [Section IV A, 32-MCT example] The text says Step II has Toffoli depth 7 and later says uncomputation (Step IV) requires an additional depth of 8; the discrepancy between 7 and 8 for the same accumulation step should be explained or corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the depth trade-off formulas are derived from the circuit construction and external cited techniques, not fitted to the claimed results.

full rationale

The paper's central results are derived from explicit circuit constructions rather than from fitting or from self-referential definitions. Theorem 1 is proved from Lemmas 4 and 5, which in turn enumerate Step II/IV depths and Step III control-count sizes for Construction 1; no parameter is fitted to the depth it then predicts. Theorem 3's ceiling(log2 n) lower bound is argued from the binary-tree information-accumulation structure of Toffoli gates and is self-contained. The citations to Khattar and Gidney, Jaques et al., Gidney, and Nie et al. are external prior works, not self-citations by the present authors, and they are used as building blocks rather than as justifications of the final lower bound. The skeptical observation that Theorem 1's formula appears inconsistent with Example 1 (depth 16 vs. 14 for n=32, m1=3) is a numerical/correctness concern about the effective Step III depth constant, not a circularity: the theorem is not equivalent to its inputs by definition. Similarly, Theorem 2 states its scope as 'in reference to Construction 1,' so any overbroad phrasing in the surrounding conclusion is an overgeneralization risk, not a circular derivation. No fitted-input-called-prediction pattern, no load-bearing self-citation, and no renaming of a known result as a new derivation were found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted constants. Its main burden is the ad hoc structural assumptions of Construction 1, especially the m2=2 split and the Step III overlap depths, which are not justified as optimal across all conditionally clean ancilla schemes.

assumptions (5)
  • domain assumption Each 2-controlled Toffoli gate combines the information of exactly two control qubits into one target, i.e., fan-in 2 for the AND accumulation.
    Used in Theorem 3's binary tree lower bound; true for the Toffoli gate but limits the bound to Clifford+Toffoli decompositions.
  • ad hoc to paper The row representation of Step II in Construction 1 correctly captures all possible conditionally clean ancilla accumulation schedules.
    Lemmas 1-4 and Theorem 1 derive exact depths from this representation; no proof exists that every conditionally clean ancilla scheme fits it.
  • ad hoc to paper For m >= 4, the optimal split is m2 = 2 ancilla for Step III and the rest m1 for Step I.
    Construction 1 asserts this split; the exact trade-off formula depends on it.
  • ad hoc to paper The effective Toffoli depth of Step III is 3 or 5 after overlapping with Steps II/IV.
    Used in Theorem 1's proof; the worked example uses 3 while the theorem uses 5, causing the inconsistency.
  • domain assumption Measurement-based uncomputation (Jaques et al. 2020) gives each Toffoli T-depth 1 with one ancilla.
    Used in Corollary 1 for the T-depth version of the lower bound.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Exact Space-Depth Trade-Offs in Multi-Controlled Toffoli Decomposition." pith.science (2026). https://pith.science/paper/K56Y5FRF

@misc{pith2026250201433,
  author       = {Pith},
  title        = {Pith review of: On Exact Space-Depth Trade-Offs in Multi-Controlled Toffoli Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K56Y5FRF}},
  note         = {Machine review of arXiv:2502.01433}
}
abstract

In this paper, we consider the optimized implementation of Multi Controlled Toffoli (MCT) using the Clifford $+$ T gate sets. While there are several recent works in this direction, here we explicitly quantify the trade-off (with concrete formulae) between the Toffoli depth (this means the depth using the classical 2-controlled Toffoli) of the $n$-controlled Toffoli (hereform we will tell $n$-MCT) and the number of clean ancilla qubits. Additionally, we achieve a reduced Toffoli depth (and consequently, T-depth), which is an extension of the technique introduced by Khattar et al. (2024). In terms of a negative result, we first show that using such conditionally clean ancilla techniques, Toffoli depth can never achieve exactly $\ceil{\log_2 n}$, though it remains of the same order. This highlights the limitation of the techniques exploiting conditionally clean ancilla [Nie et al., 2024, Khattar et al., 2024]. Then we prove that, in a more general setup, the T-Depth in the Clifford + T decomposition, via Toffoli gates, is lower bounded by $\ceil{\log_2 n}$, and this bound is achieved following the complete binary tree structure. Since the ($2$-controlled) Toffoli gate can further be decomposed using Clifford $+$ T, various methodologies are explored too in this regard for trade-off related implications.

Figures

Figures reproduced from arXiv: 2502.01433 by the authors.

Figure 1
Figure 1. FIG. 1: Equivalence between multi-controlled Pauli [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Single Toffoli decomposition with ancilla, [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. FIG. 5: Single Toffoli decomposition due to Jones [ [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figures from the paper (7 more)
Figure 3
Figure 3. Figure 3: FIG. 3: Single Toffoli decomposition without using any [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Single Toffoli decomposition due to Soeken [ [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Single Toffoli decomposition using logical-AND [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9: CCCZ circuit having T-Depth 2 [ [PITH_FULL_IMAGE:figures/full_fig_p005_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: MCT decomposition using conditionally clean [PITH_FULL_IMAGE:figures/full_fig_p006_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: 10-MCT circuit decompositions, each of them [PITH_FULL_IMAGE:figures/full_fig_p006_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: 7-MCT circuit decomposition with a Toffoli [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Transversal AND in Quantum Codes

    quant-ph 2026-03 conditional novelty 6.0 of 10

    A [[6,2,2]] qutrit code with a transversal logical AND is built from a symmetric Clifford+T circuit, and concatenation yields a [[48,2,4]] code.

Reference graph

Works this paper leans on

34 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    Assuming that the control states do not need to be restored to their original values, using the condi- tionally clean ancillae technique, the exact Toffoli depth must be strictly greater than ⌈log2 n⌉, irre- spective of the number of available ancilla

    work page

  2. [2]

    Information Security Educa- tion and Awareness (ISEA) Project phase - III, Cluster - Cryptography, initiatives of MeitY, Grant No. F.No. L-14017/1/2022-HRD

    When the control qubits are required to be returned to their original state upon completion, the Toffoli depth becomes strictly greater than 2⌈log2 n⌉. Proof. From Proposition 1, it is evident that the Toffoli depth remains constant over a range of values forn. Since here, we are interested in showing the lower bound, we consider n assumes the highest val...

    work page 2022

  3. [3]

    Wang, S., Lim, E., Li, X., Feng, J., and Chattopad- hyay, A.IFIP/IEEE 32nd International Conference on Very Large Scale Integration (VLSI-SoC), 2024

    work page 2024

  4. [4]

    IEEE Inter- national Symposium on Circuits and Systems (ISCAS), 2024

    Wang, S., Lim E., and Chattopadhyay A. IEEE Inter- national Symposium on Circuits and Systems (ISCAS), 2024

    work page 2024

  5. [5]

    Optimal Toffoli-Depth Quantum Adder

    Wang, S., Deb, S., Mondal, A., and Chattopadhyay, A. arXiv:2405.02523, May 2024

    work page Pith review arXiv 2024

  6. [6]

    ePrint Archive: 2023/286, 2023

    Chun, M., Baksi, A., and Chattopadhyay, A. ePrint Archive: 2023/286, 2023

    work page 2023

  7. [7]

    Scientific Reports, 13(1), 16338, 2023

    Wang, S., Baksi, A., and Chattopadhyay, A. Scientific Reports, 13(1), 16338, 2023

    work page 2023

  8. [8]

    IEEE Int

    Dasu, V.A., Baksi, A., Sarkar, S., and Chattopadhyay, A. IEEE Int. System-on-Chip Conference (SOCC), Sin- gapore, 2019

    work page 2019

Show all 34 references
  1. [9]

    INDOCRYPT, 2021

    Anand, R., Maitra, A., Maitra, S., Mukherjee, C.S., and Mukhopadhyay, S. INDOCRYPT, 2021

    2021

  2. [10]

    PQCrypto, 2016

    Grassl, M., Langenberg, B., Roetteler, M., and Stein- wandt, R. PQCrypto, 2016

    2016

  3. [11]

    Quantum Inf

    Bathe, B., Anand, R., and Dutta, S. Quantum Inf. Proc., 20, 394, 2021

    2021

  4. [12]

    Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2020

    Jiang, J., Sun, X., Teng, S.-H., Wu, B., Wu, K., and 13 Zhang, J. Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2020

    2020

  5. [13]

    INDOCRYPT, 2024

    Dutta, S., Ghatak, A., Chattopadhyay, A., and Maitra, S. INDOCRYPT, 2024

    2024

  6. [14]

    and Nilsson, M

    Moore, C. and Nilsson, M. SIAM Journal on Computing, 31(3): 799–815, 2001

    2001

  7. [15]

    Maslov, D. Phys. Rev. A, 93(2), 2016

    2016

  8. [16]

    41st IEEE Int

    Miller, D.M., Wille, R., and Sasanian, Z. 41st IEEE Int. Symposium on Multiple-Valued Logic, 2011

    2011

  9. [17]

    and Pedram, M

    Saeedi, M. and Pedram, M. Phys. Rev. A, 87(6), 2013

    2013

  10. [18]

    and Arusoaie, A

    Balauca, S. and Arusoaie, A. ICCS, 2022

    2022

  11. [19]

    arXiv:1904.01671, 2019

    Baker, J.M., Duckering, C., Hoover, A., and Chong, F.T. arXiv:1904.01671, 2019

    1904 arXiv

  12. [20]

    IEEE Transactions on Quantum Engineering, vol

    Paler, A., Oumarou, O., and Basmadjian, R. IEEE Transactions on Quantum Engineering, vol. 3, 2022

    2022

  13. [21]

    and Gidney, C

    Khattar, T. and Gidney, C. arXiv:2407.17966, July 2024

    2024 arXiv

  14. [22]

    Nature Communication, 15, 5886, 2024

    Claudon, B., Zylberman, J., Feniou, C., Debbasch, F., Peruzzo, A., and Piquemal, J.P. Nature Communication, 15, 5886, 2024

    2024

  15. [23]

    arXiv:2402.05053, Feb 2024

    Nie, J., Zi, W., and Sun, X. arXiv:2402.05053, Feb 2024

    2024 arXiv

  16. [24]

    using measurement-based uncomputation, which has a T-Count of 4 and a T-Depth of 2, without using any additional ancilla. When multiple logical-AND circuits are employed within the same quantum circuit, all the initial T-gates used for preparing the state |T ⟩, where |T ⟩ = TH...

    2024

  17. [25]

    and Vandaele, V

    Remaud, M. and Vandaele, V. arXiv:2501.16802, Jan 2025

    2025 arXiv

  18. [26]

    EUROCRYPT, 2020

    Jaques, S., Naehrig, M., Roetteler, M., and Virdia, F. EUROCRYPT, 2020

    2020

  19. [27]

    Quantum 2(74), 2018

    Gidney, C. Quantum 2(74), 2018

    2018

  20. [28]

    and Chuang, I.L

    Nielsen, M.A. and Chuang, I.L. Quantum computation and quantum information. Cambridge university press, 2002

    2002

  21. [29]

    IEEE Transactions on Computer-Aided Design of Inte- grated Circuits and Systems, 32(6): 818–830, 2013

    Amy, M., Maslov, D., Mosca, M., and Roetteler, M. IEEE Transactions on Computer-Aided Design of Inte- grated Circuits and Systems, 32(6): 818–830, 2013

    2013

  22. [30]

    Selinger, P. Phys. Rev. A, 87(4), 2013

    2013

  23. [31]

    Jones, C. Phys. Rev. A, 87(2): 23–28, 2013

    2013

  24. [32]

    and Jones, N.C

    Gidney, C. and Jones, N.C. arXiv:2106.11513, June 2021

    2021 arXiv

  25. [33]

    and Todo, S

    Nakanishi, K.M. and Todo, S. arXiv:2410.00910, Oct 2024

    2024 arXiv

  26. [34]

    Constructing Large Controlled Nots

    Gidney, C. Constructing Large Controlled Nots. 2015. (Part: 1–3). Accessed: November, 2024

    2015

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.