  We address the problem of estimating the phase phi given N copies of the
phase rotation gate u(phi). We consider, for the first time, the optimization
of the general case where the circuit consists of an arbitrary input state,
followed by any arrangement of the N phase rotations interspersed with
arbitrary quantum operations, and ending with a POVM. Using the polynomial
method, we show that, in all cases where the measure of quality of the estimate
phi' for phi depends only on the difference phi'-phi, the optimal scheme has a
very simple fixed form. This implies that an optimal general phase estimation
procedure can be found by just optimizing the amplitudes of the initial state.
