The Phase Matrix

Abstract

Reducing the error of quantum algorithms is often achieved by applying a primitive called amplitude amplification. Its use leads in many instances to quantum algorithms that are quadratically faster than any classical algorithm. Amplitude amplification is controlled by choosing two complex numbers φs and φt of unit norm, called phase factors. If the phases are well-chosen, amplitude amplification reduces the error of quantum algorithms, if not, it may increase the error. We give an analysis of amplitude amplification with a emphasis on the influence of the phase factors on the error of quantum algorithms. We introduce a so-called phase matrix and use it to give a straightforward and novel analysis of amplitude amplification processes. We show that we may always pick identical phase factors φs = φt with argument in the range ${{\pi}\over{3}}{\leq} {\rm arg}(\phi_{s}){\leq} {\pi}$. We also show that identical phase factors φs = φt with $-{{\pi}\over{2}}< {\rm arg}(\phi_{s})< {{\pi}\over{2}}$ never leads to an increase in the error, generalizing a recent result of Lov Grover who shows that amplitude amplification becomes a quantum analogue of classical repetition if we pick phase factors φs = φt with ${\rm arg}(\phi_{s}) = {{\pi}\over{3}}$.