Fixed-point quantum search with an optimal number of queries

TL;DR

This paper introduces a fixed-point quantum search algorithm that maintains the quadratic speedup of Grover's algorithm while requiring only a lower bound on the target state fraction, broadening practical applicability.

Contribution

It presents the first fixed-point amplitude amplification method that preserves quantum speedup without prior knowledge of the target fraction, with adjustable failure probability.

Findings

Achieves fixed-point behavior without losing quadratic advantage.

Guarantees performance over a broad range of target fractions.

Incorporates adjustable failure probability.

Abstract

Grover's quantum search and its generalization, quantum amplitude amplification, provide quadratic advantage over classical algorithms for a diverse set of tasks, but are tricky to use without knowing beforehand what fraction $\lambda$ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction, but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability, and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of $\lambda$.