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arXiv:1303.3557 · Physical Review A

Linear-depth quantum circuits for n -qubit Toffoli gates with no ancilla

Mehdi Saeedi, Massoud Pedram

Published Jun 17, 2013• 71 citations
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Abstract

We design a circuit structure with linear depth to implement an $n$-qubit Toffoli gate. The proposed construction uses a quadratic-size circuit that consists of elementary two-qubit controlled-rotation gates around the $x$ axis and uses no ancilla qubit. Circuit depth remains linear in quantum technologies with finite-distance interactions between qubits. The suggested construction is related to the long-standing construction by A. Barenco et al. [Phys. Rev. A 52, 3457 (1995)], which uses a quadratic-size, quadratic-depth quantum circuit for an $n$-qubit Toffoli gate.

DOI
10.1103/physreva.87.062318

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Comment on “Linear-depth quantum circuits for n -qubit Toffoli gates with no ancilla”

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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.

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Advanced Quantum Technologies•Dec 20, 2023•2 citations
We design a circuit structure with linear depth to implement an nnn-qubit Toffoli gate. The proposed construction uses a quadratic-size circuit that consists of elementary two-qubit controlled-rotation gates around the xxx axis and uses no ancilla qubit. Circuit depth remains linear in quantum technologies with finite-distance interactions between qubits. The suggested construction is related to the long-standing construction by A. Barenco et al. [Phys. Rev. A 52, 3457 (1995)], which uses a quadratic-size, quadratic-depth quantum circuit for an nnn-qubit Toffoli gate.
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Multiqubit quantum gates are natural quantum operations derived from various decompositions of quantum evolutions. Recently, linear-depth quantum circuits were proposed for an arbitrary nnn-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 nnn-qubit Toffoli gate. By reevaluating their circuits, we argue that their linear-depth quantum circuits could realize the joint operations of an nnn-qubit Toffoli gate and an nnn-qubit phase-flip gate.
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Decomposing (n+1)(n+1)(n+1)(n+1)‐Qubit Toffoli Gate with Shallow Circuit Depth and No 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.