Parameterized Quantum Query Complexity of Graph Collision

TL;DR

This paper introduces three new quantum algorithms for the graph collision problem, leveraging graph parameters like treewidth and a span program, and analyzes their query complexities.

Contribution

The paper presents novel quantum algorithms for graph collision based on tree decomposition, span programs, and specialized graph subclasses, improving previous bounds.

Findings

Tree decomposition-based algorithm with $O(\sqrt{n}t^{1/6})$ queries.

Span program-based algorithm with $O(\sqrt{n}+\sqrt{\alpha^{**}})$ queries.

Efficient algorithm for circulant graphs with $O(\sqrt{n})$ queries.

Abstract

We present three new quantum algorithms in the quantum query model for \textsc{graph-collision} problem: \begin{itemize} \item an algorithm based on tree decomposition that uses $O\left(\sqrt{n}t^{\sfrac{1}{6}}\right)$ queries where $t$ is the treewidth of the graph; \item an algorithm constructed on a span program that improves a result by Gavinsky and Ito. The algorithm uses $O(\sqrt{n}+\sqrt{\alpha^{**}})$ queries, where $\alpha^{**}(G)$ is a graph parameter defined by \[\alpha^{**}(G):=\min_{VC\text{-- vertex cover of}G}{\max_{\substack{I\subseteq VC\\I\text{-- independent set}}}{\sum_{v\in I}{\deg{v}}}};\] \item an algorithm for a subclass of circulant graphs that uses $O(\sqrt{n})$ queries. \end{itemize} We also present an example of a possibly difficult graph $G$ for which all the known graphs fail to solve graph collision in $O(\sqrt{n} \log^c n)$ queries.