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arXiv:2604.25861

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

Spyros Tserkis, Muhammad Umer, Eleftherios Mastorakis, Dimitris G. Angelakis

quant-phSubmitted Apr 28, 2026• Announced Apr 29, 2026• 0 citations• Revised Aug 25, 2026 (v3)
v1 · Apr 28, 2026v2 · Aug 17, 2026v3 · Aug 25, 2026
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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 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.

Comments
12 pages, 10 figures, 1 supplemental material

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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.
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 $\ceil{\log_2 n}$, 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 (222-controlled) Toffoli gate can further be decomposed using Clifford +++ T, various methodologies are explored too in this regard for trade-off related implications.
It is considered to be very important to reduce so-called the T-depth of a target quantum circuit for realizing future fault-tolerant quantum computation. Thus, there have been many researches to consider how to decompose Multiple Controlled Toffoli (MCT) gates with few T-depth; these existing methods do not consider the T-depth of each qubit when they decompose one MCT gate. In contrast to existing methods, this paper proposes to consider the T-depth of each qubit when we decompose an MCT gate; we show that we can decrease the T-depth by considering the T-depth of each qubit and appropriately selecting ancilla qubits and decomposition methods. In addition, to consider the impact of the decomposition of one MCT gate on the decomposition of subsequent gates, our method utilizes beam search to select a possibly best decomposition for each MCT gate. We confirmed that the proposed method can reduce the T-depth by an average of 17.1% compared to existing methods.
For realizing future fault-tolerant quantum computation, it is very important to reduce so-called the T-depth of a target quantum circuit. Thus, there have been many researches to consider how to decompose Multiple Controlled Toffoli (MCT) gates with few T-depth; these existing methods does not consider the T-depth of each qubit when they decompose one MCT gate. In contrast to existing methods, this paper proposes to consider the T-depth of each qubit when we decompose an MCT gate; we show that we can decrease the T-depth by considering Tdepth of each bit and appropriately selecting ancilla qubits and decomposition methods. In addition, to consider the impact of the decomposition of one MCT gate on the decomposition of subsequent gates, our method utilizes beam search to select a possibly best decomposition for each MCT gate. We confirmed that the proposed method can reduce the T-depth by an average of 17.1% compared to existing methods.
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.
The Toffoli gate is a fundamental building block for quantum arithmetic and reversible logic, yet its efficient realization remains a major challenge in both near-term and fault-tolerant quantum architectures. Recent advances in dynamic quantum circuit capabilities, including mid-circuit measurement and classical feedforward, provide new opportunities for reducing the resource overhead of non-Clifford operations. In this work, we propose a set of dynamic decomposition strategies for multi-controlled Toffoli gates that exploit adaptive circuit execution and ancilla-assisted constructions. Our methods systematically reduce entangling-gate count, T-count, and T-depth compared with conventional static decompositions, while preserving fault-tolerance guarantees. Through analytical cost models and experimental evaluation, we demonstrate that relative-phase primitives and measurement-conditioned corrections enable scalable implementations with improved depth and resource efficiency.