A Quantum Algorithm for Finding $k$-Minima

TL;DR

This paper introduces a simpler quantum algorithm for finding the $k$-minima with a query complexity of $ ext{O}(\sqrt{kN})$, applicable to various distance-based problems, and discusses a novel generalization of amplitude amplification.

Contribution

The paper presents a new, simpler quantum algorithm for $k$-minima with proven query complexity and extends amplitude amplification to find multiple solutions.

Findings

Query complexity is $ ext{O}(\sqrt{kN})$ for the proposed algorithm.

Algorithm can be adapted to $k$-nearest neighbor, clustering, and classification.

Introduces a novel generalization of amplitude amplification for multiple answers.

Abstract

We propose a new finding $k$-minima algorithm and prove that its query complexity is $\mathcal{O}(\sqrt{kN})$, where $N$ is the number of data indices. Though the complexity is equivalent to that of an existing method, the proposed is simpler. The main idea of the proposed algorithm is to search a good threshold that is near the $k$-th smallest data. Then, by using the generalization of amplitude amplification, all $k$ data are found out of order and the query complexity is $\mathcal{O}(\sqrt{kN})$. This generalization of amplitude amplification is also not well discussed and we briefly prove the query complexity. Our algorithm can be directly adapted to distance-related problems like $k$-nearest neighbor search and clustering and classification. There are few quantum algorithms that return multiple answers and they are not well discussed.