Nested Quantum Walks with Quantum Data Structures

TL;DR

This paper introduces a novel quantum walk framework utilizing quantum data structures, enabling more efficient algorithms for problems like triangle finding, and potentially leading to new upper bounds in quantum query complexity.

Contribution

It extends the quantum walk framework by incorporating quantum data structures, simplifying the construction of algorithms and reproducing key upper bounds for triangle finding.

Findings

Reproduces $O(n^{35/27})$ and $O(n^{9/7})$ upper bounds for triangle finding

Demonstrates the framework's ability to convert known bounds into algorithms

Suggests potential for discovering new bounds with easier framework

Abstract

We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data structures were considered before for searching via quantum walks. The recently proposed learning graph framework of Belovs has yielded improved upper bounds for several problems, including triangle finding and more general subgraph detection. We exhibit the power of our framework by giving a simple explicit constructions that reproduce both the $O(n^{35/27})$ and $O(n^{9/7})$ learning graph upper bounds (up to logarithmic factors) for triangle finding, and discuss how other known upper bounds in the original learning graph framework can be converted to algorithms in our framework. We hope that the ease of use of this framework will lead to the discovery of new upper bounds.