Multiqubit quantum gates are natural quantum operations derived from various decompositions of quantum evolutions. Recently, linear-depth quantum circuits were proposed for an arbitrary $n$ -qubit Toffoli gate with ancillary qubits [M. Saeedi and M. Pedram, Phys. Rev. A 87, 062318 (2013)]. Their circuits are based on an equivalent representation of the not gate regardless of the global phase. However, we find that this equivalence cannot be extended to the controlled-not gate or the $n$ -qubit Toffoli gate. By reevaluating their circuits, we argue that their linear-depth quantum circuits could realize the joint operations of an $n$ -qubit Toffoli gate and an $n$ -qubit phase-flip gate.Multiqubit quantum gates are natural quantum operations derived from various decompositions of quantum evolutions. Recently, linear-depth quantum circuits were proposed for an arbitrary $n$-qubit Toffoli gate with ancillary qubits [M. Saeedi and M. Pedram, Phys. Rev. A 87, 062318 (2013)]. Their circuits are based on an equivalent representation of the not gate regardless of the global phase. However, we find that this equivalence cannot be extended to the controlled-not gate or the $n$-qubit Toffoli gate. By reevaluating their circuits, we argue that their linear-depth quantum circuits could realize the joint operations of an $n$-qubit Toffoli gate and an $n$-qubit phase-flip gate.

Physical Review AAug 18, 20163 citations

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

Spyros Tserkis, Muhammad Umer, Eleftherios Mastorakis +1 more

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 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.

quant-pharXiv:2604.25861Apr 29, 20260 citations

Optimization of Quantum Boolean Circuits by Relative-Phase Toffoli Gates

Shohei Kuroda, Shigeru Yamashita

Lecture notes in computer scienceJan 1, 20224 citations

Decomposing (n+1) $(n+1)$ ‐Qubit Toffoli Gate with Shallow Circuit Depth and No AncillaDecomposing (n+1)$(n+1)$‐Qubit Toffoli Gate with Shallow Circuit Depth and No Ancilla

Jian Leng, Fan Yang, Xiang‐Bin Wang

Abstract Circuit depth optimization is crucial for implementing quantum algorithms in practical systems. The decomposition of ‐qubit Toffoli gate is continually optimized since this gate is a fundamental component for many quantum algorithms. Without any ancilla, ‐qubit Toffoli gate can be decomposed by standard gates with circuit depth . Here, the decomposition is further optimized and its circuit depth is reduced to . The decomposition is equivalent to ‐qubit Toffoli gate up to an overall phase and also does not need any ancilla.Abstract Circuit depth optimization is crucial for implementing quantum algorithms in practical systems. The decomposition of ‐qubit Toffoli gate is continually optimized since this gate is a fundamental component for many quantum algorithms. Without any ancilla, ‐qubit Toffoli gate can be decomposed by standard gates with circuit depth . Here, the decomposition is further optimized and its circuit depth is reduced to . The decomposition is equivalent to ‐qubit Toffoli gate up to an overall phase and also does not need any ancilla.

Advanced Quantum TechnologiesDec 20, 20232 citations

Complexity Lower Bounds for Quantum Circuits

Mika Hirvensalo

Springer eBooksJan 1, 20010 citations

Multiqubit quantum gates are natural quantum operations derived from various decompositions of quantum evolutions. Recently, linear-depth quantum circuits were proposed for an arbitrary

n

-qubit Toffoli gate with ancillary qubits [M. Saeedi and M. Pedram, Phys. Rev. A 87, 062318 (2013)]. Their circuits are based on an equivalent representation of the not gate regardless of the global phase. However, we find that this equivalence cannot be extended to the controlled-not gate or the

n

-qubit Toffoli gate. By reevaluating their circuits, we argue that their linear-depth quantum circuits could realize the joint operations of an

n

-qubit Toffoli gate and an

n

-qubit phase-flip gate.

Decomposing (n+1)

(n+1)

‐Qubit Toffoli Gate with Shallow Circuit Depth and No Ancilla