Abstract
We generalize quantum circuits for the Toffoli gate presented by Selinger [1] and Jones [2] 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 [3] and Welch et al [4]. 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.
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