  In a previous paper we have presented a general scheme for the implementation
of symmetric generalized measurements (POVMs) on a quantum computer. This
scheme is based on representation theory of groups and methods to decompose
matrices that intertwine two representations. We extend this scheme in such a
way that the measurement is minimally disturbing, i.e., it changes the state
vector \ket{\Psi} of the system to \sqrt{\Pi} \ket{\Psi} where \Pi is the
positive operator corresponding to the measured result.
  Using this method, we construct quantum circuits for measurements with
Heisenberg-Weyl symmetry. A continuous generalization leads to a scheme for
optimal simultaneous measurements of position and momentum of a Schr"odinger
particle moving in one dimension such that the outcomes satisfy \Delta x \Delta
p \geq \hbar.
  The particle to be measured collides with two probe particles, one for the
position and the other for the momentum measurement. The position and momentum
resolution can be tuned by the entangled joint state of the probe particles
which is also generated by a collision with hard-core potential. The parameters
of the POVM can then be controlled by the initial widths of the wave functions
of the probe particles. We point out some formal similarities and differences
to simultaneous measurements of quadrature amplitudes in quantum optics.
