Pith. sign in

REVIEW 1 major objections 2 minor 60 references

Teleportation enables unit Toffoli depth for multi-controlled gates regardless of control count.

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 Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-05-07 16:36 UTC

load-bearing objection Teleportation gets arbitrary MCT to Toffoli depth 1, but the correction schedule is the part that still needs explicit verification. the 1 major comments →

arxiv 2604.25861 v1 submitted 2026-04-28 quant-ph

Minimum Toffoli depth for the multi-controlled Toffoli gate via teleportation

Spyros Tserkis , Muhammad Umer , Eleftherios Mastorakis , Dimitris G. Angelakis This is my paper
classification quant-ph
keywords multi-controlled ToffoliToffoli depthteleportationquantum circuit decompositionancilla qubitsquantum algorithmsMCT gate
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.

This paper establishes that the multi-controlled Toffoli gate can be realized with a fixed Toffoli depth of one using a teleportation approach. The decomposition works for any number of controls by relying on pre-shared entanglement and additional ancilla qubits. This keeps the total Toffoli count low relative to other methods. As a result, quantum circuits for tasks like addition, memory access, and decision trees gain shallower implementations.

Core claim

The central discovery is a teleportation-based decomposition that implements an arbitrary multi-controlled Toffoli gate with unit Toffoli depth, independent of the number of controls, while maintaining a relatively low Toffoli count compared to existing approaches at the cost of linear ancilla overhead and distributed entangled pairs.

What carries the argument

A teleportation-based decomposition that uses pre-distributed entangled pairs and ancilla qubits to achieve unit Toffoli depth for the MCT gate.

Load-bearing premise

Entangled pairs can be distributed across distant qubits without adding significant depth or error.

What would settle it

Implementing the decomposition for an MCT gate with five or more controls on a quantum processor supporting entanglement distribution and verifying that the sequential Toffoli layer count remains exactly one.

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

If this is right

  • The adder operator can be built with Toffoli depth independent of control number.
  • Quantum read-only memory circuits become shallower in depth.
  • Quantum neurons and decision trees achieve reduced overall circuit depth.
  • Any quantum algorithm relying on MCT gates inherits the constant-depth property.

Where Pith is reading between the lines

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

  • Hardware platforms already equipped for entanglement distribution stand to gain immediate depth reductions in MCT-heavy algorithms.
  • Similar teleportation tricks may apply to depth reduction for other families of multi-controlled gates.
  • The linear ancilla cost versus depth trade-off points to hardware connectivity as a key design factor for scaling.

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

1 major / 2 minor

Summary. The manuscript presents a teleportation-based decomposition of the multi-controlled Toffoli (MCT) gate that achieves a Toffoli depth of exactly 1 independent of the number of controls. The construction uses a linear number of ancilla qubits together with distributed Bell pairs; the resulting Toffoli count is stated to be competitive with prior decompositions. The method is illustrated on several MCT-heavy circuits including adders, quantum read-only memory, quantum neurons, and decision trees.

Significance. A verified constant-depth MCT primitive would be useful for depth-sensitive quantum algorithms, since conventional MCT decompositions incur depth linear in the number of controls. The teleportation approach trades ancilla count and entanglement distribution (already demonstrated on several platforms) for depth reduction, which is a plausible engineering trade-off. The applications section provides concrete examples that could be re-analyzed with the new primitive.

major comments (1)
  1. [teleportation construction] The central unit-depth claim requires that every Toffoli—including all Pauli corrections arising from the teleportation measurements—can be scheduled into a single parallel layer with no data dependencies. The manuscript must supply an explicit correction circuit or timing diagram (with gate ordering) to confirm that the depth bound is preserved; without it the scheduling assumption remains unverified.
minor comments (2)
  1. [Abstract] The abstract asserts a 'relatively low Toffoli count' but supplies neither the exact asymptotic expression nor a side-by-side comparison table with the best known prior decompositions; adding both would make the resource claim precise.
  2. [applications] The applications (adder, QROM, etc.) would be strengthened by explicit before/after Toffoli-depth tables rather than qualitative statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback. We address the single major comment below and will revise the manuscript to provide the requested explicit verification.

read point-by-point responses

  1. Referee: The central unit-depth claim requires that every Toffoli—including all Pauli corrections arising from the teleportation measurements—can be scheduled into a single parallel layer with no data dependencies. The manuscript must supply an explicit correction circuit or timing diagram (with gate ordering) to confirm that the depth bound is preserved; without it the scheduling assumption remains unverified.

    Authors: We agree that an explicit timing diagram would strengthen the presentation and remove any ambiguity. The Pauli corrections arising from the teleportation measurements are single-qubit X and Z gates; they are not Toffoli gates and therefore do not contribute to Toffoli depth. In the teleportation construction these corrections act on the target and control registers after the single parallel Toffoli layer and can be applied concurrently because they commute with the preceding operations and have no data dependencies on one another. The overall Toffoli depth therefore remains exactly one. To make the scheduling fully transparent we will add a detailed circuit diagram (including gate ordering and parallelization of the corrections) to the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: construction uses standard teleportation primitives without self-referential definitions or fitted inputs

full rationale

The paper presents a teleportation-based circuit construction for the MCT gate. No equations reduce the claimed unit Toffoli depth to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors. The derivation relies on distributing Bell pairs and applying Pauli corrections in parallel, which are external primitives. The central result is a new scheduling of known operations rather than a renaming or self-definition. This matches the default expectation of a non-circular constructive result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard quantum teleportation and entanglement distribution without introducing new free parameters or invented entities.

axioms (1)
  • domain assumption Quantum teleportation can be used to implement controlled gate operations when entangled pairs can be distributed between qubits.
    Invoked to achieve constant depth independent of control count.

pith-pipeline@v0.9.0 · 5467 in / 1118 out tokens · 53381 ms · 2026-05-07T16:36:27.875988+00:00 · methodology

0 comments
read the original abstract

The decomposition of complex quantum operations into experimentally feasible gate sets has been a central challenge since the early development of quantum computing. The multi-controlled Toffoli (MCT) gate is a key example, with applications across a wide range of quantum algorithms, whose decomposition into smaller gates, however, typically leads to deep circuits. In this work, we introduce a teleportation-based decomposition that implements an arbitrary MCT gate with unit Toffoli depth, independent of the number of controls, while maintaining a relatively low Toffoli count compared to existing approaches. This is achieved at the cost of a linear overhead in ancilla qubits and the ability to distribute entangled pairs across distant qubits, a capability already available in several quantum computing platforms. We further demonstrate the advantages of this implementation in circuits that rely on MCT gates, such as the adder operator, quantum read-only memory, quantum neurons, and quantum decision trees.

Figures

Figures reproduced from arXiv: 2604.25861 by Dimitris G. Angelakis, Eleftherios Mastorakis, Muhammad Umer, Spyros Tserkis.

Figure 1
Figure 1. Figure 1: FIG. 1. Quantum circuit representation of an view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. In panels (a)-(c) we plot the Toffoli depth, the Toffoli count, and the ancilla count, respectively, against the number of control qubits in view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. An MCT view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Protocol for the teleportation of an MCT view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Teleportation-based decomposition of an MCT view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Dutta view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Panels (a) and (b) depict side by side the upper bound of process fidelity of the Dutta view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Panel (a) depicts the MCT-based construction of the adder view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The Toffoli depth of the adder operator is plotted for three view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Illustrations of MCT gates across diverse quantum circuit ar view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    Vedral, A

    V . Vedral, A. Barenco, and A. Ekert, Quantum networks for elementary arithmetic operations, Phys. Rev. A54, 147 (1996)

    work page 1996

  2. [2]

    B. L. Douglas and J. B. Wang, Efficient quantum circuit imple- mentation of quantum walks, Phys. Rev. A79, 052335 (2009)

    work page 2009

  3. [3]

    Lubasch, J

    M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Vari- ational quantum algorithms for nonlinear problems, Phys. Rev. A101, 010301 (2020)

    work page 2020

  4. [4]

    Jaksch, P

    D. Jaksch, P. Givi, A. J. Daley, and T. Rung, Variational quan- tum algorithms for computational fluid dynamics, AIAA J.61, 1885 (2023)

    work page 2023

  5. [5]

    M. Umer, E. Mastorakis, and D. G. Angelakis, Efficient estima- tion and sequential optimization of cost functions in variational quantum algorithms, Quantum Sci. and Technol.10, 035022 (2025)

    work page 2025

  6. [6]

    Tomesh, N

    T. Tomesh, N. Allen, D. Dilley, and Z. Saleem, Quantum- classical tradeoffs and multi-controlled quantum gate decom- positions in variational algorithms, Quantum8, 1493 (2024)

    work page 2024

  7. [7]

    P. Over, S. Bengoechea, T. Rung, F. Clerici, L. Scandurra, E. de Villiers, and D. Jaksch, Boundary treatment for variational quantum simulations of partial differential equations on quan- tum computers, Comput. Fluids.288, 106508 (2025)

    work page 2025

  8. [8]

    Phalak, M

    K. Phalak, M. Alam, A. Ash-Saki, R. O. Topaloglu, and S. Ghosh, Optimization of quantum read-only memory circuits, arXiv:2204.03097 (2022)

    work page arXiv 2022

  9. [9]

    Tacchino, C

    F. Tacchino, C. Macchiavello, D. Gerace, and D. Bajoni, An artificial neuron implemented on an actual quantum processor, NPJ Quantum Inf.5, 1 (2019)

    work page 2019

  10. [10]

    Cuellar, L

    M. Cuellar, L. Ruiz, and M. Pegalajar, Implementation of clas- sical decision trees in a quantum computing paradigm, inInter- national Conference on Hybrid Artificial Intelligence Systems (Springer, 2024) pp. 226–237

    work page 2024

  11. [11]

    Balauca and A

    S. Balauca and A. Arusoaie, Efficient constructions for simu- lating multi controlled quantum gates, inComputational Sci- ence – ICCS 2022(Springer International Publishing, 2022) p. 179–194

    work page 2022

  12. [12]

    N. A. Nemkov, E. O. Kiktenko, I. A. Luchnikov, and A. K. Fe- dorov, Efficient variational synthesis of quantum circuits with coherent multi-start optimization, Quantum7, 993 (2023)

    work page 2023

  13. [13]

    K. M. Nakanishi, T. Satoh, and S. Todo, Decompositions of multiple controlled-zgates on various qubit-coupling graphs, Phys. Rev. A110, 012604 (2024)

    work page 2024

  14. [14]

    J. Nie, W. Zi, and X. Sun, Quantum circuit for multi-qubit Tof- foli gate with optimal resource, arXiv:2402.05053 (2024)

    work page arXiv 2024

  15. [15]

    A. Kole, A. Deb, K. Datta, and R. Drechsler, Dynamic realiza- tion of multiple control Toffoli gate, in2024 Design, Automa- tion & Test in Europe Conference & Exhi- bition (DATE)(IEEE, 2024) p. 1–6

    work page 2024

  16. [16]

    Heußen, D

    S. Heußen, D. F. Locher, and M. M ¨uller, Measurement-free fault-tolerant quantum error correction in near-term devices, PRX Quantum5, 010333 (2024)

    work page 2024

  17. [17]

    Khattar and C

    T. Khattar and C. Gidney, Rise of conditionally clean ancillae for efficient quantum circuit constructions, Quantum9, 1752 (2025)

    work page 2025

  18. [18]

    Dutta, S

    S. Dutta, S. Wang, A. Baksi, A. Chattopadhyay, and S. Maitra, Exact space-depth trade-offs in multicontrolled Toffoli decom- position, Phys. Rev. A111, 052611 (2025)

    work page 2025

  19. [19]

    Zindorf and S

    B. Zindorf and S. Bose, Efficient implementation of multicon- trolled quantum gates, Phys. Rev. Appl.24(2025)

    work page 2025

  20. [20]

    Zindorf and S

    B. Zindorf and S. Bose, All you need is pi: Quantum computing with hermitian gates, Quantum9, 1925 (2025)

    work page 1925

  21. [21]

    Moore and M

    C. Moore and M. Nilsson, Parallel quantum computation and quantum codes, SIAM J. Comput.31, 799–815 (2001)

    work page 2001

  22. [22]

    Anders, D

    J. Anders, D. K. L. Oi, E. Kashefi, D. E. Browne, and E. An- dersson, Ancilla-driven universal quantum computation, Phys. Rev. A82(2010)

    work page 2010

  23. [23]

    T. J. Proctor, E. Andersson, and V . Kendon, Universal quantum computation by the unitary control of ancilla qubits and using a fixed ancilla-register interaction, Phys. Rev. A88(2013)

    work page 2013

  24. [24]

    A. B. Watts, R. Kothari, L. Schaeffer, and A. Tal, Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits, inProceedings of the 51st An- nual ACM SIGACT Symposium on Theory of Computing, STOC ’19 (ACM, 2019) p. 515–526

    work page 2019

  25. [25]

    Liu and A

    Z. Liu and A. Gheorghiu, Depth-efficient proofs of quantum- ness, Quantum6, 807 (2022)

    work page 2022

  26. [26]

    Buhrman, M

    H. Buhrman, M. Folkertsma, B. Loff, and N. M. P. Neumann, State preparation by shallow circuits using feed forward, Quan- tum8, 1552 (2024)

    work page 2024

  27. [27]

    Y . Quek, E. Kaur, and M. M. Wilde, Multivariate trace estima- tion in constant quantum depth, Quantum8, 1220 (2024)

    work page 2024

  28. [28]

    B ¨aumer, V

    E. B ¨aumer, V . Tripathi, D. S. Wang, P. Rall, E. H. Chen, S. Ma- jumder, A. Seif, and Z. K. Minev, Efficient long-range entangle- ment using dynamic circuits, PRX Quantum5, 030339 (2024)

    work page 2024

  29. [29]

    B ¨aumer and S

    E. B ¨aumer and S. Woerner, Measurement-based long-range en- tangling gates in constant depth, Phys. Rev. Res.7, 023120 (2025)

    work page 2025

  30. [30]

    Cao and J

    C. Cao and J. Eisert, Measurement-driven quantum advantages in shallow circuits, Phys. Rev. Lett.136, 080601 (2026)

    work page 2026

  31. [31]

    Tserkis, M

    S. Tserkis, M. Umer, and D. G. Angelakis, Depth optimization of CNOT ladder circuits, arXiv:2511.13256 (2025)

    work page arXiv 2025

  32. [32]

    Sarvaghad-Moghaddam and M

    M. Sarvaghad-Moghaddam and M. Zomorodi, A general pro- tocol for distributed quantum gates, Quantum Inf. Process.20 (2021)

    work page 2021

  33. [33]

    Pompili, S

    M. Pompili, S. L. N. Hermans, S. Baier, H. K. C. Beukers, P. C. Humphreys, R. N. Schouten, R. F. L. Vermeulen, M. J. Tiggelman, L. dos Santos Martins, B. Dirkse, S. Wehner, and R. Hanson, Realization of a multinode quantum network of re- mote solid-state qubits, Science372, 259–264 (2021)

    work page 2021

  34. [34]

    Storz, J

    S. Storz, J. Sch¨ar, A. Kulikov, P. Magnard, P. Kurpiers, J. L¨utolf, T. Walter, A. Copetudo, K. Reuer, A. Akin, J.-C. Besse, M. Gabureac, G. J. Norris, A. Rosario, F. Martin, J. Martinez, W. Amaya, M. W. Mitchell, C. Abellan, J.-D. Bancal, N. San- gouard, B. Royer, A. Blais, and A. Wallraff, Loophole-free Bell inequality violation with superconducting ci...

    work page 2023

  35. [35]

    Li and J

    Y . Li and J. D. Thompson, High-rate and high-fidelity modular interconnects between neutral atom quantum processors, PRX Quantum5, 020363 (2024)

    work page 2024

  36. [36]

    C. M. Knaut, A. Suleymanzade, Y .-C. Wei, D. R. Assump- cao, P.-J. Stas, Y . Q. Huan, B. Machielse, E. N. Knall, M. Su- tula, G. Baranes, N. Sinclair, C. De-Eknamkul, D. S. Levonian, M. K. Bhaskar, H. Park, M. Lon ˇcar, and M. D. Lukin, Entan- glement of nanophotonic quantum memory nodes in a telecom network, Nature629, 573–578 (2024)

    work page 2024

  37. [37]

    O’Reilly, G

    J. O’Reilly, G. Toh, I. Goetting, S. Saha, M. Shalaev, A. L. Carter, A. Risinger, A. Kalakuntla, T. Li, A. Verma, and C. Monroe, Fast photon-mediated entanglement of continu- ously cooled trapped ions for quantum networking, Phys. Rev. Lett.133, 090802 (2024)

    work page 2024

  38. [38]

    PsiQuantum Team, A manufacturable platform for photonic quantum computing, Nature641, 876–883 (2025)

    work page 2025

  39. [39]

    S. Saha, M. Shalaev, J. O’Reilly, I. Goetting, G. Toh, 10 A. Kalakuntla, Y . Yu, and C. Monroe, High-fidelity remote en- tanglement of trapped atoms mediated by time-bin photons, Nat. Commun.16(2025)

    work page 2025

  40. [40]

    H. Song, X. Zhang, L. Komza, N. Fiaschi, Y . Xiong, Y . Zhi, S. Dhuey, A. Schwartzberg, T. Schenkel, G. Hautier, Z.-H. Zhang, and A. Sipahigil, Entanglement of a nuclear spin qubit register in silicon photonics, Nat. Nanotechnol.21, 53–57 (2025)

    work page 2025

  41. [41]

    M. A. Nielsen and I. L. Chuang, Programmable quantum gate arrays, Phys. Rev. Lett.79, 321 (1997)

    work page 1997

  42. [42]

    Gottesman and I

    D. Gottesman and I. L. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single- qubit operations, Nature402, 390–393 (1999)

    work page 1999

  43. [43]

    Eisert, K

    J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Plenio, Optimal local implementation of nonlocal quantum gates, Phys. Rev. A 62, 052317 (2000)

    work page 2000

  44. [44]

    P. M. Q. Cruz and B. Murta, Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging, APL Quantum1(2024)

    work page 2024

  45. [45]

    Georgopoulos, C

    K. Georgopoulos, C. Emary, and P. Zuliani, Modeling and sim- ulating the noisy behavior of near-term quantum computers, Phys. Rev. A104, 062432 (2021)

    work page 2021

  46. [46]

    Treinish, Qiskit/qiskit-metapackage: Qiskit 0.44.0 (2023)

    M. Treinish, Qiskit/qiskit-metapackage: Qiskit 0.44.0 (2023)

    work page 2023

  47. [47]

    Raginsky, A fidelity measure for quantum channels, Phys

    M. Raginsky, A fidelity measure for quantum channels, Phys. Lett. A290, 11–18 (2001)

    work page 2001

  48. [48]

    Gilchrist, N

    A. Gilchrist, N. K. Langford, and M. A. Nielsen, Distance mea- sures to compare real and ideal quantum processes, Phys. Rev. A71(2005)

    work page 2005

  49. [49]

    H. F. Hofmann, Complementary classical fidelities as an ef- ficient criterion for the evaluation of experimentally realized quantum operations, Phys. Rev. Lett.94(2005)

    work page 2005

  50. [50]

    Comp.C–28, 267–268 (1979)

    Kunz, On the equivalence between one-dimensional discrete Walsh-Hadamard and multidimensional discrete Fourier trans- forms, IEEE Trans. Comp.C–28, 267–268 (1979)

    work page 1979

  51. [51]

    Ekert and R

    A. Ekert and R. Jozsa, Quantum computation and Shor’s factor- ing algorithm, Rev. Mod. Phys.68, 733 (1996)

    work page 1996

  52. [52]

    Georgopoulos, C

    K. Georgopoulos, C. Emary, and P. Zuliani, Comparison of quantum-walk implementations on noisy intermediate-scale quantum computers, Phys. Rev. A103, 022408 (2021)

    work page 2021

  53. [53]

    Nandi, S

    B. Nandi, S. Singha, A. Datta, A. Saha, and A. Chakrabarti, Robust implementation of discrete-time quantum walks in any finite-dimensional quantum system, Mod. Phys. Lett. A40, 2550035 (2025)

    work page 2025

  54. [54]

    A. F. Shaw, P. Lougovski, J. R. Stryker, and N. Wiebe, Quan- tum Algorithms for Simulating the Lattice Schwinger Model, Quantum4, 306 (2020)

    work page 2020

  55. [55]

    J. Li, S. Lin, K. Yu, and G. Guo, Quantum k-nearest neighbor classification algorithm based on Hamming distance, Quantum Inf. Process.21, 18 (2022)

    work page 2022

  56. [56]

    Beverland, E

    M. Beverland, E. Campbell, M. Howard, and V . Kliuchnikov, Lower bounds on the non-Clifford resources for quantum com- putations, Quantum Sci. and Technol.5, 035009 (2020)

    work page 2020

  57. [57]

    Babbush, C

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spectra in quantum circuits with linear T complexity, Phys. Rev. X8, 041015 (2018)

    work page 2018

  58. [58]

    S ¨underhauf, E

    C. S ¨underhauf, E. Campbell, and J. Camps, Block-encoding structured matrices for data input in quantum computing, Quan- tum8, 1226 (2024). 11 Supplemental Material: Minimum Toffoli depth for the multi-controlled Toffoli gate via teleportation Spyros Tserkis,1, 2 Muhammad Umer,3 Eleftherios Mastorakis,1 and Dimitris G. Angelakis 2, 3, 4 1School of Electr...

    work page 2024

  59. [59]

    van den Berg, Z

    E. van den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors, Nat. Physics19, 1116–1121 (2023)

    work page 2023

  60. [60]

    B ¨aumer, V

    E. B ¨aumer, V . Tripathi, D. S. Wang, P. Rall, E. H. Chen, S. Majumder, A. Seif, and Z. K. Minev, Efficient long-range entanglement using dynamic circuits, PRX Quantum5, 030339 (2024). S-1 (a) (b) (c) FIG. S-1. Panels (a) and (b) depict side by side the upper bound of process fidelity of the Duttaet al.decomposition and the teleportation-based decomposit...

    work page 2024