Optimal quantum searching to find a common element of two sets

TL;DR

This paper introduces a variant of Grover's quantum search algorithm that optimally finds a common element in two sets, outperforming the straightforward approach by a constant factor in certain cases.

Contribution

It proposes a new quantum search algorithm variant that achieves optimal performance for finding common elements between two sets.

Findings

Achieves optimal query complexity in specific cases

Reduces the number of oracle queries compared to standard Grover's algorithm

Provides theoretical analysis of the algorithm's performance

Abstract

Given two sets A and B and two oracles O(A) and O(B) that can identify the elements of these sets respectively, the goal is to find an element common to both sets using minimum number of oracle queries. Each application of either O(A) or O(B) is taken as a single oracle query. This is basically a search problem and a straightforward application of Grover algorithm can solve this problem but its performance is slow compared to the optimal one by a constant factor of 1.57. Here we present a variant of Grover algorithm which achieves the optimal performance in not too restrictive cases.