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arXiv:2502.01433

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

Dutta, Suman, Siyi Wang, Anubhab Baksi, Anupam Chattopadhyay, Subhamoy Maitra

Published Feb 3, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2502.01433

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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 nnn-controlled Toffoli (hereform we will tell nnn-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 \ceillog⁡2n\ceil{\log_2 n}\ceillog2​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 \ceillog⁡2n\ceil{\log_2 n}\ceillog2​n, and this bound is achieved following the complete binary tree structure. Since the (222-controlled) Toffoli gate can further be decomposed using Clifford +++ T, various methodologies are explored too in this regard for trade-off related implications.
In this paper we consider the optimized implementation of multicontrolled Toffoli decomposition using the Clifford+T\text{Clifford}+TClifford+T gate set. While there are several recent works in this direction, here we explicitly quantify the trade-off (with concrete formulas) between the Toffoli depth (this means the depth using the classical 2-controlled Toffoli gate) of the nnn-controlled Toffoli decomposition and the number of clean ancilla qubits. Additionally, we achieve a reduced Toffoli (consequently TTT) depth, which is an extension of the technique introduced by Khattar and Gidney [T. Khattar and C. Gidney, arXiv:2407.17966]. In terms of a negative result, we first show that by using such conditionally clean ancilla techniques, the Toffoli depth can never achieve exactly $\ensuremath{\lceil}{log}_{2}n\ensuremath{\rceil}$, though it remains of the same order. This highlights the limitation of the techniques exploiting conditionally clean ancillas [J. Nie et al., arXiv:2402.05053; T. Khattar and C. Gidney, arXiv:2407.17966]. Then we prove that, in a more general setup, the TTT depth in the Clifford+T\text{Clifford}+TClifford+T decomposition, via Toffoli gates, is lower bounded by $\ensuremath{\lceil}{log}_{2}n\ensuremath{\rceil}$, and this bound is achieved following the complete binary tree structure. Since the (2-controlled) Toffoli gate can further be decomposed using Clifford+T\text{Clifford}+TClifford+T gate set, various methodologies are also explored in this regard for trade-off-related implications.
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.
Efficient Synthesis of Multi-Controlled Toffoli Gates with Ternary Clifford+P9+P_9+P9​ Gates
Temporary occupation of qutrit levels can reduce the width and depth required for binary circuit logic in quantum computing. We introduce an efficient intermediate-qutrit decomposition of multi-controlled Toffoli gates with binary-subspace inputs and outputs. For balanced control widths n=2h−1n=2^h-1n=2h−1, the proposed decomposition is evaluated hierarchically through a tree, allowing independent subtree computations to proceed in parallel and reducing the depth from linear to logarithmic. The balanced low-depth construction requires 6n+36n+36n+3 logical P9P_9P9​ injections and n−34\frac{n-3}{4}4n−3​ clean ancillary qutrits. It matches the direct P9P_9P9​ count of a recursively extended Clifford+P9P_9P9​ baseline derived from the state-of-the-art work while using asymptotically one-quarter as many clean ancillas and replacing linear depth with logarithmic depth. For arbitrary control widths, a sequential extension preserves the 6n+36n+36n+3 P9P_9P9​ count but has depth O(log⁡m+n−m)O(\log m+n-m)O(logm+n−m) for a balanced core m=2h−1≤nm=2^h-1\leq nm=2h−1≤n, which is linear in the worst case over nnn. By reducing the resource overhead of a widely used reversible primitive, the proposed decomposition provides a practical building block for the design and compilation of more resource-efficient quantum algorithms in the fault-tolerant regime.
Abstract One of the key operations in various quantum algorithms, including Grover’s algorithm, is the multi-controlled Toffoli (MCT) gate. This work proposes optimal design techniques for the MCT gate using the conditionally clean ancillae strategy, assuming that an arbitrary number of clean ancillae are available. We consider two primary cases: first, when only a single clean ancilla is available; and second, when the number of clean ancillae is O(n)\mathcal {O}(n)O(n) O ( n ) , where n is the number of controls in the MCT gate. For the case of a single clean ancilla, we propose a novel design method that achieves better efficiency compared to a previous approach. While prior work yields a Toffoli-depth of O(n)\mathcal {O}(n)O(n) O ( n ) , our proposed method reduces the Toffoli-depth to O(n)\mathcal {O}(\sqrt{n})O(n​) O ( n ) . Moreover, by employing this new technique as a subroutine within previously proposed methods for the cases of two or three clean ancillae, we demonstrate both logically and empirically an improvement in time complexity. Additionally, prior work addressed optimal designs only when the number of clean ancillae was up to approximately log⁡n\log nlogn log n , without providing efficient strategies beyond this regime. In this study, we also present efficient MCT gate constructions when a sufficiently large number of clean ancillae are available. We confirm, both theoretically and empirically, that the Toffoli-depth decreases as the number of clean ancillae increases. In conclusion, this work highlights that the optimal MCT design strategy depends on the availability of clean ancillae. It also indirectly suggests a threshold point at which the optimal design strategy transitions, depending on the number of ancillae provided.
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 MCT gate. This decomposition has unit Toffoli depth, independent of the number of controls, while achieving the lowest Toffoli count among existing approaches for more than five controls at the cost of a linear overhead in ancilla qubits. We also investigate the parameter regimes in which the teleportation-based decomposition is preferable by analyzing the relative quality of distributed entanglement and Toffoli gates. 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.
The multi-controlled Toffoli (MCT) gate is a key primitive in quantum arithmetic, oracle construction, and quantum cryptanalysis. Although recent work has established optimal Toffoli-depth MCT decompositions under all-to-all qubit connectivity, their realization on near-term quantum hardware with restricted qubit connectivity remains largely unexplored. While general-purpose quantum mappers can route arbitrary circuits, they do not explicitly exploit the repeated interaction patterns inherent in MCT decompositions. In our present paper, we study architecture-aware mappings of optimal Toffoli-depth MCT decompositions onto restricted two-dimensional qubit layouts. We begin with a structured geometric placements that preserve the parallelism of state-of-the-art Toffoli and MCT decompositions with no additional depth overhead. We further introduce a motif-based packing framework in which decomposition layers are represented by interaction motifs derived from basic Toffoli gates. By embedding these motifs vertex-disjointly into hardware graphs, we characterize the minimum-size topologies supporting the required qubit resources and derive explicit bounds on the resulting depth overhead under tight qubit budgets. Finally, we compare these bounds with routing-aware placement heuristics and empirically evaluate the effectiveness of embedding different motifs across a range of hardware topologies.