Fixed-point quantum continuous search algorithm with optimal query complexity

TL;DR

This paper introduces a fixed-point quantum search algorithm tailored for continuous search problems, achieving quadratic speedup and establishing optimal query complexity bounds, with broad applicability to optimization and eigenvalue problems.

Contribution

The work presents a novel quantum search algorithm for continuous domains, demonstrating its optimality and providing a framework for problem-specific oracle design.

Findings

Achieves quadratic speedup over classical methods.

Establishes a lower bound on quantum query complexity for CSPs.

Provides a general framework for applying the algorithm to various problems.

Abstract

Continuous search problems (CSPs), which involve finding solutions within a continuous domain, frequently arise in fields such as optimization, physics, and engineering. Unlike discrete search problems, CSPs require navigating an uncountably infinite space, presenting unique computational challenges. In this work, we propose a fixed-point quantum search algorithm that leverages continuous variables to address these challenges, achieving a quadratic speedup. Inspired by the discrete search results, we manage to establish a lower bound on the query complexity of arbitrary quantum search for CSPs, demonstrating the optimality of our approach. In addition, we demonstrate how to design the internal structure of the quantum search oracle for specific problems. Furthermore, we develop a general framework to apply this algorithm to a range of problem types, including optimization and eigenvalue problems involving continuous variables.