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Erschienen in:
2013 | OriginalPaper | Buchkapitel
Grover’s Algorithm with Errors
verfasst von : Andris Ambainis, Artūrs Bačkurs, Nikolajs Nahimovs, Alexander Rivosh
Erschienen in: Mathematical and Engineering Methods in Computer Science
Verlag: Springer Berlin Heidelberg
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Grover’s algorithm is a quantum search algorithm solving the unstructured search problem of size
n
in
$O(\sqrt{n})$
queries, while any classical algorithm needs
O
(
n
) queries [3].
However, if query has some small probability of failing (reporting that none of the elements are marked), then quantum speed-up disappears: no quantum algorithm can be faster than a classical exhaustive search by more than a constant factor [8].
We study the behaviour of Grover’s algorithm in the model there query may report
some
marked elements as unmarked (each marked element has its own error probability, independent of other marked elements).
We analyse the limiting behaviour of Grover’s algorithm for a large number of steps and prove the existence of limiting state
ρ
lim
. Interestingly, the limiting state is independent of error probabilities of individual marked elements. If we measure
ρ
lim
, the probability of getting one of the marked states
i
1
, …,
i
k
is
$\frac{k}{k+1}$
. We show that convergence time is
O
(
n
).