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Erschienen in:
2010 | OriginalPaper | Buchkapitel
Quantum Counterfeit Coin Problems
verfasst von : Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, Junichi Teruyama
Erschienen in: Algorithms and Computation
Verlag: Springer Berlin Heidelberg
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The counterfeit coin problem requires us to find all false coins from a given bunch of coins using a balance scale. We assume that the balance scale gives us only “balanced” or “tilted” information and that we know the number
k
of false coins in advance. The balance scale can be modeled by a certain type of oracle and its query complexity is a measure for the cost of weighing algorithms (the number of weighings). In this paper, we study the quantum query complexity for this problem. Let
Q
(
k
,
N
) be the quantum query complexity of finding all
k
false coins from the
N
given coins. We show that for any
k
and
N
such that
k
<
N
/2,
Q
(
k
,
N
) =
O
(
k
1/4
), contrasting with the classical query complexity, Ω(
k
log(
N
/
k
)), that depends on
N
. So our quantum algorithm achieves a
quartic
speed-up for this problem. We do not have a matching lower bound, but we show some evidence that the upper bound is tight: any algorithm, including our algorithm, that satisfies certain properties needs Ω(
k
1/4
) queries.