How to construct a CCY gate in Qiskit

Question

How can one construct a CCY gate using gates which are native to Qiskit (CCX and single qubit gates). I was able to find the answer for CCZ gates, however guessing and testing until I can figure out CCY seems like a bad way to go.

Answer 1

For $\mathrm{CC} Y$, given that you have a decomposition for $\mathrm{CC} Z$ (or for $\textrm{CC} X = \mathrm{Toffoli}$), all you need is the relationship between $ X$, $Y$, and $Z$: $$ Y = R_z (-\tfrac\pi2) X R_z (\tfrac\pi2) = R_x (\tfrac\pi2) Z R_x(-\tfrac\pi2) $$ Then, given $Y = U P U^\dagger$ for some $P$ for which you know a decomposition for $\mathrm{CC} P$, simply do the analogous decomposition: $$ \mathrm{CC}Y = (\mathbf 1 \otimes \mathbf 1 \otimes U) \mathrm{CC} P (\mathbf 1 \otimes \mathbf 1 \otimes U)^\dagger. $$ You can do this for any self-inverse unitary $V$ in place of $Y$ as well.