Fourier Analytic Approach to Quantum Estimation of Group Action

This article proposes a unified method to estimation of group action by using the inverse Fourier transform of the input state. The method provides optimal estimation for commutative and non-commutative group with/without energy constraint. The proposed method can be applied to projective representations of non-compact groups as well as of compact groups. This paper addresses the optimal estimation of R, U(1), SU(2), SO(3), and R^2 with Heisenberg representation under a suitable energy constraint.