## Designing $(n+1)$-gates using 1-gates and Toffoli gates

Let $U$ be an $(n+1)$-gate. Assume that $U = \big(a_{ij}\big)$ is the matrix representation of $U$ with respect to the computational basis. The goal of this post is to prove that $U$ can be designed using only Toffoli gates and gates of the form $(I_{n} \otimes A)$, for various 1-gates $A$. By this post, we…