We generalize quantum circuits for the Toffoli gate presented by Selinger and Jones for functionally controlled NOT gates, i.e., $X$ gates controlled by arbitrary $n$-variable Boolean functions. Our constructions target the gate set consisting of Clifford gates and single qubit rotations by arbitrary angles. Our constructions use the Walsh-Hadamard spectrum of Boolean functions and build on the work by Schuch and Siewert and Welch et al. We present quantum circuits for the case where the target qubit is in an arbitrary state as well as the special case where the target is in a known state. Additionally, we present constructions that require no auxiliary qubits and constructions that have a rotation depth of 1.

中文翻译：

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功能控制非门的量子电路

我们概括了由 Selinger 和 Jones 提出的用于功能控制非门的 Toffoli 门的量子电路，即由任意 $n$ 变量布尔函数控制的 $X$ 门。我们的结构针对由 Clifford 门和任意角度的单个量子位旋转组成的门组。我们的构造使用布尔函数的 Walsh-Hadamard 谱，并建立在 Schuch 和 Siewert 以及 Welch 等人的工作之上。我们为目标量子位处于任意状态的情况以及目标处于已知状态的特殊情况提供了量子电路。此外，我们提出了不需要辅助量子位的结构和旋转深度为 1 的结构。