  We consider group-covariant positive operator valued measures (POVMs) on a
finite dimensional quantum system. Following Neumark's theorem a POVM can be
implemented by an orthogonal measurement on a larger system. Accordingly, our
goal is to find an implementation of a given group-covariant POVM by a quantum
circuit using its symmetry. Based on representation theory of the symmetry
group we develop a general approach for the implementation of group-covariant
POVMs which consist of rank-one operators. The construction relies on a method
to decompose matrices that intertwine two representations of a finite group. We
give several examples for which the resulting quantum circuits are efficient.
In particular, we obtain efficient quantum circuits for a class of POVMs
generated by Weyl-Heisenberg groups. These circuits allow to implement an
approximative simultaneous measurement of the position and crystal momentum of
a particle moving on a cyclic chain.
