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

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.