$$\sum_{j=0}^{m-1}  \left(
\begin{array}{c} 1 \\ \omega^j \end{array} \right)
(1, \omega^{-j}) = \sum_{j=0}^{m-1} \left(
\begin{array}{cc}1&\omega^{-j}\\ \omega^{j}&1 \end{array} \right) = m
 I_2$$ the elements ${|\Psi_j\rangle}= \sqrt{1/m}(1,\omega^{j-1} )^T \in
{\mathbb C}^2$ for $j \in \{1, \ldots, m\}$ define a POVM with $m=n$ operators on a qubit. Therefore, the unitary matrix $\tilde{M} \in
{\mathbb C}^{m \times m}$ is a unitary extension of the matrix $$M=\sqrt{\frac{1}{m}} \left( \begin{array}{cccc}
1&1&\ldots&1\\1&\omega& \ldots & \omega^{m-1} \end{array} \right) \in
{\mathbb C}^{2 \times m}.$$ $M$ corresponds to the first two rows of the discrete Fourier matrix $$F_m=\sqrt{\frac{1}{m}}\left(\omega^{jk}\right)_{j,k=0}^{m-1}
\in {\mathbb C}^{m \times m}$$ of size $m$. Consequently, by considering the qubit to be measured as a subsystem of an $m$-dimensional system the implementation of the inverse Fourier transform ${\tilde M}^\dagger = F_m^\dagger$ leads to the probability distribution of the cyclic POVM on the qubit.

For the construction of a quantum circuit we have to embed the system of dimension $m$ into a register with $l$ qubits. The register must have $r := 2^l \geq m$ dimensions. Following Section \[Sektion ortho\] we extend the cyclic POVM by an appropriate number of zero operators $A_{m+1}, \ldots , A_{r} = 0_2 \in {\mathbb C}^{2 \times
2}$. We therefore have $$M = \sqrt{\frac{1}{m}}\left(
\begin{array}{cccc|ccc} 1 