POVM three-qubit circuit for symmetric quantum states

Question

I have been reading this paper but don't yet understand how to implement a circuit to determine in which state the qubit is not for a cyclic POVM. More specifically, I want to implement a cyclic POVM with $m=3$;

Update: I came across having to implement this unitary matrix: $$ M= \frac{1}{\sqrt{2}}\left[ {\begin{array}{cc} 1 & 1 \\ 1 & w \\ \end{array} } \right] $$ Where $w$ is a third root of unity using rotations, after which I am stuck.

Answer 1

This is not the unitary that you have to implement: you need a two-qubit unitary $$ \frac{1}{\sqrt{3}}\left(\begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & \omega & \omega^2 & 0 \\ 1 & \omega^2 & \omega & 0 \\ 0 & 0 & 0 & \sqrt{3} \end{array}\right), $$ where $\omega=e^{2i\pi/3}$, the point being that if you introduce an ancilla qubit in the state 0, apply this unitary, and then measure in the computational basis, the 3 measurement outcomes 00, 01 and 10 correspond to the 3 POVM elements.

I don't (yet) have a circuit implementation for this. You'll see the paper you cite carefully avoids talking about the Fourier transform in non-power of 2 dimensions. You certainly could use the standard constructions based on Givens rotations, but the result is going to be fairly horrible.

Here's an attempt at a circuit. I've made a few tweaks since I last ran it through a computer to check, so it's always possible that a slight error has crept in, but broadly... Here, I'm using $Z^r$ to denote $$ \left(\begin{array}{cc} 1 & 0 \\ 0 & e^{i\pi r} \end{array}\right), $$ and $$ V=\frac{1}{\sqrt{3}}\left(\begin{array}{cc} 1 & \sqrt{2} \\ -\sqrt{2} & 1 \end{array}\right). $$

Answer 2

Check this
$$\frac{1}{\sqrt{3}}\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & \omega & \omega^2 \\ 1 & \omega^2 & \omega \\ \end{array}\right) = \left(\begin{array}{cc} H & 0 \\ 0 & 1 \\ \end{array}\right) \cdot \frac{1}{\sqrt{3}}\left(\begin{array}{ccc} \sqrt{2} & 0 & 1 \\ 0 & \sqrt{3} & 0 \\ 1 & 0 & -\sqrt{2} \\ \end{array}\right) \cdot M_{3} $$

$$M_{3} = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}}i\omega^2 & \frac{1}{\sqrt{2}}i\omega \\ 0 & -\frac{1}{\sqrt{2}}\omega^2 & -\frac{1}{\sqrt{2}}\omega \\ \end{array}\right) $$

$$ \frac{1}{\sqrt{2}}\left(\begin{array}{ccc} i\omega^2 & i\omega \\ -\omega^2 & -\omega \\ \end{array}\right) = X \cdot S \cdot X \cdot Z \cdot H \cdot \left(\begin{array}{ccc} \omega^2 & 0 \\ 0 & \omega \\ \end{array}\right) \cdot Z $$