Optimal Quantum Circuits for General Phase Estimation
Wim van Dam, Giacomo Mauro D’Ariano, ARTUR K. EKERT, Chiara Macchiavello, Michele Mosca · Physical Review Letters · 2007
We address the problem of estimating the phase $\ensuremath{\phi}$ given $N$ copies of the phase-rotation gate ${u}_{\ensuremath{\phi}}$. 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 general measurement. Using the polynomial method, we show that, in all cases where the measure of quality of the estimate $\stackrel{\texttildelow{}}{\ensuremath{\phi}}$ for $\ensuremath{\phi}$ depends only on the difference $\stackrel{\texttildelow{}}{\ensuremath{\phi}}\ensuremath{-}\ensuremath{\phi}$, 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.