# Doing maths with controlled-half NOTs

In Quantum Computation with the simplest maths possible there is a section titled "Doing maths with a controlled-half NOT" which covers a reversible-(N)AND circuit with controlled-half NOTs.

• What would the unitary matrix for a controlled-half NOT be?

• How could a reversible-XNOR gate be constructed with controlled-half NOTs?

This is the gate that I would call controlled-square-root-of-not. Bit more of a mouthful, I know, but perhaps conveys more accurately what it's doing. The point is that it's a unitary $U$ such that $U^2$ is the controlled-not. There are probably a few ways of writing down such a thing, but, for example $$U=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{e^{i\pi/4}}{\sqrt{2}} & \frac{e^{-i\pi/4}}{\sqrt{2}} \\ 0 & 0 & \frac{e^{-i\pi/4}}{\sqrt{2}} & \frac{e^{i\pi/4}}{\sqrt{2}} \end{array}\right)$$