Möbius Transformations in Quantum Amplitude Amplification With Generalized Phases

Abstract

The iteration of the operators employed in quantum amplitude ampli¯cation with generalized phases is analyzed by using elementary properties (geometric and algebraic) of the M€ obius transformations (fractional linear transformations). It is shown that, for a given quantum algorithm without measurement, which produces a good state with probability a of success, if the phase angles ' and which mark the good and initial states respectively satisfy ' ¼ with a small enough, then, for a number n of iterations with n 2 Âð1= ffiffiffi a p Þ we get an error probability that is at most Oða 2 Þ.