of [@Nielsen] to obtain a quantum circuit for ${\tilde
M}^\dagger$. The algorithm decomposes the matrix ${\tilde M}^\dagger$ into a product of two-level matrices that can be translated into a sequence of elementary gates, i.e., each gate operates on one or two qubits. In general, the constructed circuit for ${\tilde M}^\dagger$ is of exponential size in the number $n$ of POVM operators. Intuitively, some symmetry properties of the considered POVMs may lead to algorithms constructing smaller circuits than the standard algorithm that works for arbitrary unitary matrices. To specify the symmetry of POVMs on a single qubit using geometric concepts, we use the correspondence of POVMs to points on the Bloch sphere as already mentioned in the introduction. Usually, each point on the Bloch sphere is considered as a pure state. Specifically, a pure state $\rho \in{\mathbb C}^{2\times 2}$ corresponds to the point $(x,y,z)^T \in {\mathbb R}^3$ on the Bloch sphere with $$\left(
\begin{array}{c}x\\y\\z \end{array} \right) = \left( 
\begin{array}{c} {\rm tr}(\sigma_x \rho ) \\ {\rm tr}
(\sigma_y \rho) \\ {\rm tr}( \sigma_z \rho) \end{array} \right)$$ where $$\sigma_x=\left( \begin{array}{cc}0&1\\1&0\end{array}\right),
\sigma_y=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right), \; {\rm
and} \; \sigma_z=\left( \begin{array}{cc}1&0\\0&-1\end{array}
\right)$$ denote the Pauli spin matrices. Conversely, the point $(x,y,z)^T \in {\mathbb R}^3$ on the Bloch 