Quantum Searching Amidst Uncertainty

Abstract

Consider a database most of whose entries are marked but the precise fraction of marked entries is not known. What is known is that the fraction of marked entries is 1–ε, where ε is a random variable that is uniformly distributed in the range (0,ε ₀).The problem is to try to select a marked item from the database in a single query. If the algorithm selects a marked item, it succeeds, else if it selects an unmarked item, it makes an error.

How low can we make the probability of error? The best possible classical algorithm can lower the probability of error to O(ε $_{\rm 0}^{\rm 2}$). The best known quantum algorithms for this problem could also only lower the probability of error to O( ε $_{\rm 0}^{\rm 2}$). Using a recently invented quantum search technique, this paper gives an algorithm that reduces the probability of error to O(ε $_{\rm 0}^{\rm 3}$). The algorithm is asymptotically optimal.