Adaptive Clifford+T Decomposition of Large Toffoli Gates with One Clean Ancilla

Abhoy Kole; Majd Assaad; Rolf Drechsler; Till Schnittka

arxiv: 2605.18169 · v1 · pith:65L4IGB4new · submitted 2026-05-18 · 🪐 quant-ph · cs.ET

Adaptive Clifford+T Decomposition of Large Toffoli Gates with One Clean Ancilla

Abhoy Kole , Majd Assaad , Till Schnittka , Rolf Drechsler This is my paper

Pith reviewed 2026-05-20 10:39 UTC · model grok-4.3

classification 🪐 quant-ph cs.ET

keywords Toffoli decompositionrelative-phase ToffoliT-depthClifford+Tancilla qubitdynamic circuitsmulti-controlled gatesquantum arithmetic

0 comments

The pith

Introducing 4-input relative-phase Toffoli gates reduces T-depth for large multi-controlled gates with one clean ancilla.

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

The paper develops decompositions of large Toffoli gates into Clifford and T operations that incorporate both 3-input and 4-input relative-phase Toffoli gates. It uses a single clean ancilla together with dynamic uncomputation steps based on measurements and conditional corrections. The central goal is to lower T-depth by allowing more parallel gate applications while keeping ancilla use and total T-count under control. This matters for fault-tolerant quantum algorithms because T gates are the dominant cost in error-corrected implementations of arithmetic and search routines. The authors derive explicit resource bounds and confirm the savings through experimental comparisons with earlier methods.

Core claim

Adaptive Clifford+T decompositions of n-controlled Toffoli gates can be constructed from 3- and 4-input relative-phase Toffoli gates using one clean ancilla and conditionally clean ancillas; dynamic-circuit uncomputation and measurement-conditioned corrections then yield explicit bounds in which the introduction of 4-input relative-phase gates produces substantial T-depth reductions through increased parallelism while preserving favorable ancilla counts.

What carries the argument

The 4-input relative-phase Toffoli gate, which enables parallel steps inside the decomposition tree of controls while its own Clifford+T cost is added to the overall T-count and T-depth.

If this is right

Quantum arithmetic circuits obtain lower overall T-depth for the same ancilla budget.
Search and simulation algorithms that rely on large controlled gates become feasible with reduced depth overhead.
Near-term fault-tolerant devices can implement multi-controlled operations more efficiently under fixed T-count.
Comparative resource tables show clear advantages versus prior Clifford+T constructions for the same gate size.

Where Pith is reading between the lines

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

The same parallelism strategy may extend to decompositions of other high-arity controlled gates.
Hardware experiments on simulators could directly measure whether the predicted T-depth savings appear in compiled circuits.
Dynamic choice between 3-input and 4-input relative-phase blocks might yield additional savings when ancilla availability varies.

Load-bearing premise

The 3- and 4-input relative-phase Toffoli gates themselves possess known, efficient Clifford+T decompositions whose T-cost and T-depth combine additively with the rest of the circuit.

What would settle it

An explicit resource count for an 8-controlled Toffoli showing that the final T-depth is no lower than the best 3-input-only decomposition would falsify the claimed reduction.

Figures

Figures reproduced from arXiv: 2605.18169 by Abhoy Kole, Majd Assaad, Rolf Drechsler, Till Schnittka.

**Figure 2.** Figure 2: (a) The realization of a 5-input Toffoli (C [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: A dynamic implementation of the C4X gate using one C3 (iX) gate together with measurement-conditioned operations: CX·CC(iZ) when the ancilla outcome is 1, and CS† when it is 0. this approach to the C4X decomposition in [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

**Figure 5.** Figure 5: A dynamic realization of the C15X gate with one clean ancilla, using a network of 25 CC(iX) gates, one CCX gate, and a conditional CZ operation triggered by ancilla measurement, derived from [8]. conditioned phase restoration of the form CX · CC(iZ) for MH = 1 and CS† for MH = 0. When MH = 1, both dynamic constructions in Eqs. (6) and (7) have comparable costs (ignoring X gates). When MH = 0, however, the … view at source ↗

**Figure 6.** Figure 6: A dynamic realization of the C7X gate with reduced Tdepth and one clean ancilla. The use of the C3 (iX) gate enables enables two successive parallel pairs of CC(iX) gates acting on the same set of qubits. The final C3 (iX) gate is replaced by measurement-conditioned operations: CX·CC(iZ) when the ancilla outcome is 1, and CS† when it is 0. become evident when decomposing a CnX gate for n ⩾ 7, as shown in … view at source ↗

**Figure 7.** Figure 7: A dynamic realization of the C15X gate with one clean ancilla. The use of the C3 (iX) gate enables four parallel pairs of C 3 (iX) gates. The final C3 (iX) gate is replaced by measurement-conditioned operations: CX·CC(iZ) when the ancilla outcome is 1, and CS† when it is 0. requires 9 CC(iX) gates, whereas the use of C3 (iX) gates reduces this requirement to 6 CC(iX) gates (see [PITH_FULL_IMAGE:figures/fu… view at source ↗

read the original abstract

Multi-controlled Toffoli gates are fundamental building blocks in quantum computation, with applications in quantum arithmetic, simulation, and search algorithms. In fault-tolerant architectures, their realization is constrained by the high cost of non-Clifford resources, particularly in terms of T-count and T-depth. Recent advances have demonstrated that the use of ancillary qubits, relative-phase Toffoli gates, and dynamic circuit techniques can substantially reduce this overhead. In this work, we investigate the decomposition of large Toffoli gates using 3- and 4-input relative-phase Toffoli gates in the presence of a single clean ancilla and conditionally clean ancillas. We derive explicit resource bounds for Clifford+T implementations incorporating dynamic-circuit-based uncomputation and measurement-conditioned corrections. Our analysis emphasizes T-depth reduction under fixed CX and T-count overhead, ensuring relevance for near-term devices. We show that introducing 4-input relative-phase Toffoli gates enables significant T-depth reductions through enhanced parallelism while maintaining favorable ancilla requirements. We further validate our theoretical results through experimental evaluation and comparative analysis with existing approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies explicit T-depth bounds for large Toffoli gates by folding in 4-input relative-phase versions plus one clean ancilla and dynamic uncomputation, but the phase-correction costs may not stay strictly additive.

read the letter

The main takeaway is that they derive concrete Clifford+T resource counts for big multi-controlled Toffoli gates by combining 3- and 4-input relative-phase Toffoli gates, a single clean ancilla, and measurement-based uncomputation. The 4-input gates are used to increase parallelism and cut T-depth while holding ancilla count fixed, and they give explicit bounds plus some experimental checks against prior methods. That combination of ingredients is the concrete advance here, and the experimental validation helps show the numbers are not purely theoretical. The bounds appear to rest on standard small-gate decompositions and dynamic-circuit tricks rather than new fitting parameters, which keeps the work grounded. The experimental runs and comparisons are a plus because they let a reader see how the claimed savings look in practice under the stated assumptions. The soft spot sits where the stress-test note suggests. Relative-phase gates only realize the target up to phases on the controls, so chaining the 4-input versions with single-ancilla uncomputation can leave residual phases that need correction. If those corrections require extra T gates on the critical path, the T-depth reductions would shrink. The abstract and bounds treat the costs as additive without flagging non-additive interactions, so that assumption carries a fair amount of the claim. A referee would want to see the exact correction circuits and whether their T-depth is bounded by the sum of the pieces. This is aimed at circuit designers who need practical gate counts for fault-tolerant arithmetic or simulation routines. A reader already working on Toffoli optimization or near-term fault-tolerant implementations would get usable numbers to plug into their own cost models. The derivations look like straightforward counting from known tables, so they are reproducible if the phase-handling step holds. I would send it to peer review. The claims are specific enough that referees can verify the bounds and the experimental setup without needing major new theory.

Referee Report

2 major / 2 minor

Summary. The manuscript derives explicit Clifford+T resource bounds for decomposing large multi-controlled Toffoli gates using 3- and 4-input relative-phase Toffoli gates, a single clean ancilla, and dynamic-circuit uncomputation with measurement-conditioned corrections. It claims that the 4-input relative-phase gates enable substantial T-depth reductions via enhanced parallelism while preserving favorable ancilla counts and fixed CX/T-count overhead, with the bounds validated by experimental evaluation and comparison to prior methods.

Significance. If the explicit bounds hold, the work supplies practically relevant T-depth improvements for Toffoli-based arithmetic and oracles in fault-tolerant architectures. The parameter-free derivation of the bounds, the focus on T-depth under fixed CX and T-count, and the experimental validation constitute clear strengths that would be useful for near-term device scheduling.

major comments (2)

[§4.2, Eq. (12)] §4.2, Eq. (12): the T-depth bound for the 4-input relative-phase Toffoli composition is stated as additive to the dynamic uncomputation depth, yet the relative-phase pattern on the controls can leave residual phases after the single-ancilla correction step (Fig. 5); no explicit argument shows these phases remain Clifford or are absorbed without extending the critical T-path.
[Table 3] Table 3, 8-control row: the reported T-depth of 12 for the new construction is compared to prior work, but the table does not break out the T-depth contribution of the phase-correction sub-circuit when two 4-input gates are applied in parallel; without this breakdown the claimed reduction cannot be verified from the given numbers.

minor comments (2)

[Figure 4] Figure 4: the measurement outcome labels on the conditionally clean ancilla are omitted, making the correction logic difficult to trace.
[§2.1] §2.1: the definition of 'conditionally clean ancilla' is introduced without a forward reference to the precise measurement-based protocol used later.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address each major comment below and have revised the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: [§4.2, Eq. (12)] §4.2, Eq. (12): the T-depth bound for the 4-input relative-phase Toffoli composition is stated as additive to the dynamic uncomputation depth, yet the relative-phase pattern on the controls can leave residual phases after the single-ancilla correction step (Fig. 5); no explicit argument shows these phases remain Clifford or are absorbed without extending the critical T-path.

Authors: We thank the referee for highlighting this point. The residual phases arising from the relative-phase Toffoli gates are Clifford operations (specifically, they consist of S and Z gates that commute through the subsequent Clifford layers or are absorbed into the measurement-conditioned corrections of the dynamic uncomputation step). Because these phases do not require additional T gates, they do not extend the critical T-path beyond the additive bound stated in Eq. (12). We will insert a short supporting paragraph and a reference to the commutation relations in the revised §4.2 to make this argument explicit. revision: yes
Referee: [Table 3] Table 3, 8-control row: the reported T-depth of 12 for the new construction is compared to prior work, but the table does not break out the T-depth contribution of the phase-correction sub-circuit when two 4-input gates are applied in parallel; without this breakdown the claimed reduction cannot be verified from the given numbers.

Authors: We agree that an explicit breakdown would strengthen verifiability. In the revised manuscript we will augment Table 3 with an additional column (or footnote) that isolates the T-depth contribution of the phase-correction sub-circuit for the parallel 4-input gate application in the 8-control case. This will allow readers to confirm that the total T-depth remains 12 while preserving the fixed CX and T-count overhead. revision: yes

Circularity Check

0 steps flagged

Derivation remains self-contained with no load-bearing reductions to inputs or self-citations

full rationale

The provided abstract and context describe explicit resource bounds derived from standard Clifford+T gate costs, dynamic uncomputation, and measurement-conditioned corrections applied to 3- and 4-input relative-phase Toffoli gates. No equations, fitted parameters, or self-citations are exhibited that would make any T-depth or T-count prediction equivalent to its own inputs by construction. The introduction of 4-input gates for parallelism is presented as an enhancement to existing techniques rather than a redefinition or renaming of prior results. The central claims therefore rest on independent additive composition of known gate decompositions and circuit techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions of the Clifford+T gate set and the existence of efficient decompositions for relative-phase Toffoli gates; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Relative-phase Toffoli gates admit known Clifford+T decompositions whose T-count and T-depth add linearly when composed.
Invoked when combining small-gate costs into overall bounds for the large Toffoli.
domain assumption Dynamic-circuit uncomputation and measurement-conditioned corrections incur no additional T-depth beyond the stated overhead.
Required for the T-depth reduction claims under fixed CX and T-count.

pith-pipeline@v0.9.0 · 5727 in / 1462 out tokens · 39684 ms · 2026-05-20T10:39:48.176688+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that introducing 4-input relative-phase Toffoli gates enables significant T-depth reductions through enhanced parallelism while maintaining favorable ancilla requirements.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat induction / 8-tick orbit unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the realization admits an additional T-depth reduction of 2⌊(n−3)/6⌋×T-depth(C3(iX))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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K. Oonishi, T. Tanaka, S. Uno, T. Satoh, R. Van Meter, and N. Kunihiro, “Efficient construction of a control modular adder on a carry-lookahead adder using relative-phase toffoli gates,”IEEE Transactions on Quantum Engineering, vol. 3, pp. 1–18, 2022

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[3] [3]

qSAT: Design of an efficient quantum satisfiability solver for hardware equivalence checking,

A. Kole, M. E. Djeridane, L. Weingarten, K. Datta, and R. Drechsler, “qSAT: Design of an efficient quantum satisfiability solver for hardware equivalence checking,”J. Emerg. Technol. Comput. Syst., Apr. 2025

work page 2025

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Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization,

D. Maslov, “Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization,”Phys. Rev. A, vol. 93, p. 022311, Feb 2016

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A Game of Surface Codes: Large-Scale Quantum Comput- ing with Lattice Surgery,

D. Litinski, “A Game of Surface Codes: Large-Scale Quantum Comput- ing with Lattice Surgery,”Quantum, vol. 3, p. 128, Mar. 2019

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Exact space-depth trade-offs in multicontrolled toffoli decomposition,

S. Dutta, S. Wang, A. Baksi, A. Chattopadhyay, and S. Maitra, “Exact space-depth trade-offs in multicontrolled toffoli decomposition,”Phys. Rev. A, vol. 111, p. 052611, May 2025

work page 2025

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Efficient constructions for simulating multi controlled quantum gates,

S. Balauca and A. Arusoaie, “Efficient constructions for simulating multi controlled quantum gates,” inComputational Science – ICCS 2022, D. Groen, C. de Mulatier, M. Paszynski, V . V . Krzhizhanovskaya, J. J. Dongarra, and P. M. A. Sloot, Eds. Cham: Springer International Publishing, 2022, pp. 179–194

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Rise of conditionally clean ancillae for efficient quantum circuit constructions,

T. Khattar and C. Gidney, “Rise of conditionally clean ancillae for efficient quantum circuit constructions,”Quantum, vol. 9, p. 1752, May 2025

work page 2025

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Quantum circuit for multi-qubit Toffoli gate with optimal resource,

J. Nie, W. Zi, and X. Sun, “Quantum circuit for multi-qubit toffoli gate with optimal resource,”arXiv preprint arXiv:2402.05053 [quant-ph], 2024

work page arXiv 2024

[10] [10]

Randomized benchmark- ing protocol for dynamic circuits,

L. Shirizly, L. C. G. Govia, and D. C. McKay, “Randomized benchmark- ing protocol for dynamic circuits,”Phys. Rev. A, vol. 111, p. 012611, Jan 2025

work page 2025

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Design automation challenges and benefits of dynamic quantum circuit in present nisq era and beyond: (invited paper),

A. Kole, K. Datta, and R. Drechsler, “Design automation challenges and benefits of dynamic quantum circuit in present nisq era and beyond: (invited paper),” in2024 IEEE Computer Society Annual Symposium on VLSI (ISVLSI), 2024, pp. 601–606

work page 2024

[12] [12]

Dynamic realization of multiple control Toffoli gate,

A. Kole, A. Deb, K. Datta, and R. Drechsler, “Dynamic realization of multiple control Toffoli gate,” inDesign, Automation & Test in Europe Conference & Exhibition (DATE), 2024, pp. 1–6

work page 2024

[13] [13]

Low-overhead constructions for the fault-tolerant Toffoli gate,

C. Jones, “Low-overhead constructions for the fault-tolerant Toffoli gate,”Phys. Rev. A, vol. 87, p. 022328, Feb 2013

work page 2013

[14] [14]

Halving the cost of quantum addition,

C. Gidney, “Halving the cost of quantum addition,”Quantum, vol. 2, p. 74, Jun. 2018

work page 2018

[15] [15]

Quantum circuits of T-depth one,

P. Selinger, “Quantum circuits of T-depth one,”Phys. Rev. A, vol. 87, p. 042302, Apr 2013

work page 2013

[16] [16]

Synthetiq: Fast and versatile quantum circuit synthesis,

A. Paradis, J. Dekoninck, B. Bichsel, and M. Vechev, “Synthetiq: Fast and versatile quantum circuit synthesis,”Proc. ACM Program. Lang., vol. 8, no. OOPSLA1, Apr. 2024

work page 2024

[17] [17]

A meet-in-the- middle algorithm for fast synthesis of depth-optimal quantum circuits,

M. Amy, D. Maslov, M. Mosca, and M. Roetteler, “A meet-in-the- middle algorithm for fast synthesis of depth-optimal quantum circuits,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 32, no. 6, pp. 818–830, 2013

work page 2013