Quantum Counterfeit Coin Problems

TL;DR

This paper investigates the quantum query complexity of the counterfeit coin problem, demonstrating a quartic speed-up over classical methods for finding false coins when their number is less than half of the total.

Contribution

It introduces a quantum algorithm that significantly reduces the number of queries needed to identify false coins compared to classical approaches.

Findings

Quantum algorithm achieves O(k^{1/4}) query complexity.

Classical complexity is (k \u2212 extlog(N/k)).

Evidence suggests the quantum upper bound is nearly tight.

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 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 \Omega(k^{1/4}) queries.