Decomposing Quantum Generalized Toffoli with an Arbitrary Number of Ancilla

We present a general decomposition of the Generalized Toffoli, and for completeness, the multi-target gate using an arbitrary number of clean or dirty ancilla. While prior work has shown how to decompose the Generalized Toffoli using 0, 1, or $O(n)$ many clean ancilla and 0, 1, and $n-2$ dirty ancilla, we provide a generalized algorithm to bridge the gap, i.e. this work gives an algorithm to generate a decomposition for any number of clean or dirty ancilla. While it is hard to guarantee optimality, our decompositions guarantee a decrease in circuit depth as the number of ancilla increases.