Decoding Quantum Search Advantage: The Critical Role of State Properties in Random Walks

TL;DR

This paper investigates how quantum state properties like coherence and entanglement influence the success of quantum random-walk search algorithms, providing analytical insights into their role in quantum speedups and guiding future algorithm design.

Contribution

It introduces three variants of quantum random-walk search algorithms with exact success probability formulas linked to state properties, revealing how coherence and entanglement affect performance.

Findings

Increased coherence fraction improves success probability.

Greater entanglement and coherence can reduce success probability.

Algorithms achieve Grover-like speedups and have potential for quantum machine learning.

Abstract

Quantum algorithms have demonstrated provable speedups over classical counterparts, yet establishing a comprehensive theoretical framework to understand the quantum advantage remains a core challenge. In this work, we decode the quantum search advantage by investigating the critical role of quantum state properties in random-walk-based algorithms. We propose three distinct variants of quantum random-walk search algorithms and derive exact analytical expressions for their success probabilities. These probabilities are fundamentally determined by specific initial state properties: the coherence fraction governs the first algorithm's performance, while entanglement and coherence dominate the outcomes of the second and third algorithms, respectively. We show that increased coherence fraction enhances success probability, but greater entanglement and coherence reduce it in the latter two cases. These findings reveal fundamental insights into harnessing quantum properties for advantage and guide algorithm design. Our searches achieve Grover-like speedups and show significant potential for quantum-enhanced machine learning.