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In a previous paper we have presented a general scheme for the implementationof symmetric generalized measurements (POVMs) on a quantum computer. Thisscheme is based on representation theory of groups and methods to decomposematrices that intertwine two representations. We extend this scheme in such away that the measurement is minimally disturbing, i.e., it changes the statevector \ket{\Psi} of the system to \sqrt{\Pi} \ket{\Psi} where \Pi is thepositive operator corresponding to the measured result.  Using this method, we construct quantum circuits for measurements withHeisenberg-Weyl symmetry. A continuous generalization leads to a scheme foroptimal simultaneous measurements of position and momentum of a Schr"odingerparticle moving in one dimension such that the outcomes satisfy \Delta x \Deltap \geq \hbar.  The particle to be measured collides with two probe particles, one for theposition and the other for the momentum measurement. The position and momentumresolution can be tuned by the entangled joint state of the probe particleswhich is also generated by a collision with hard-core potential. The parametersof the POVM can then be controlled by the initial widths of the wave functionsof the probe particles. We point out some formal similarities and differencesto simultaneous measurements of quadrature amplitudes in quantum optics.Comment: 32 pages, latex, 6 figures, presentation improved, The implementation  scheme for the continuous-variable measurement has been added. To appear in  Journ. Math. Phy

Minimally disturbing Heisenberg–Weyl symmetric measurements using hard-core collisions of Schrödinger particles