& 1 & \ldots & 1 & 0 & \ldots & 0 \\ 1 &
\omega & \ldots & \omega^{m-1} & 0 & \ldots & 0 \end{array} \right)
\in {\mathbb C}^{2 \times r}.$$ A possible unitary extension ${\tilde
M}$ of this matrix is given by ${\tilde M} = F_m \oplus I_{r-m} \in
{\mathbb C}^{r \times r}$ where $I_{r-m}$ denotes the identity matrix of size $r-m$. Consequently, on a qubit register the cyclic POVM corresponding to the $m$-sided regular polygon can be implemented by performing the operation ${\tilde M}^\dagger = F_m^\dagger \oplus I_{r
- m}$. The circuit for implementing the cyclic POVM is schematically shown in Figure \[Figur DFT\]. Note that the embedding $\rho \mapsto
\rho \oplus 0_{r-m}$ corresponds to the use of initialized ancilla qubits.

The Fourier transform can be implemented efficiently if $m$ is a power of two [@Nielsen]. Furthermore, the embedding into a qubit register is straightforward since we do not need zero operators in this case. In summary, the cyclic POVM can be implemented efficiently on a qubit register if $m$ is a power of two. For instance, the quantum circuit for the implementation of the cyclic POVM is shown in Figure \[Figur DFT\] for $m=4$. The 