2005 | OriginalPaper | Buchkapitel
The Phase Matrix
verfasst von : Peter Høyer
Erschienen in: Algorithms and Computation
Verlag: Springer Berlin Heidelberg
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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
φ
s
and
φ
t
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
φ
s
=
φ
t
with argument in the range
${{\pi}\over{3}}{\leq} {\rm arg}(\phi_{s}){\leq} {\pi}$
. We also show that identical phase factors
φ
s
=
φ
t
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
φ
s
=
φ
t
with
${\rm arg}(\phi_{s}) = {{\pi}\over{3}}$
.