set $\Omega$ of outcomes. Furthermore, the considered POVMs have the following properties:

1.  The family $(A_j)$ describes a single-qubit measurement, i.e., the system Hilbert space is $\C^2$.

2.  Each $A_j$ is a rank-one operator, i.e., $A_j={|\Psi_j\rangle}{\langle \Psi_j|}$. The vectors ${|\Psi_j\rangle}$ have the same length. They are not necessarily normalized.

3.  The operators $A_j$ correspond to symmetric points on the Bloch sphere. The symmetry groups are finite subgroups of $SO(3)$. The possible symmetry groups are the cyclic and dihedral groups and the symmetry groups of the platonic solids.

These properties show that we restrict our attention to a rather specific class of symmetric POVMs. The symmetry is fundamental in our constructions of the circuits implementing the POVMs. Specifically, we choose a cyclic subgroup of the symmetry group corresponding to a POVM. Under the action of the cyclic group the set of points on the Bloch sphere decomposes into several orbits. As shown in Section \[Section 2.1\] POVMs given by a single orbit can easily be implemented by a discrete Fourier transform. Since we have several orbits we have to use additional gates besides the Fourier transform to implement the POVM. This explains why the discrete Fourier transform plays a central role 