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 a quantum circuit implementation of a given group-covariant POVM
which uses the symmetry of the POVM. Based on representation theory of the
symmetry group we develop a general approach for the implementation of groupcovariant
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.