Quantum Counterfeit Coin Problems
Abstract
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, \Omega(k\log(N/k)), that depends on N. So our quantum algorithm achieves a quartic speedup 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 \Omega(k^{1/4}) queries.
 Publication:

arXiv eprints
 Pub Date:
 September 2010
 arXiv:
 arXiv:1009.0416
 Bibcode:
 2010arXiv1009.0416I
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics
 EPrint:
 18 pages