Quantum query complexity of minor-closed graph properties

TL;DR

This paper investigates the quantum query complexity of minor-closed graph properties, revealing that most such properties require on the order of n^{3/2} queries, with some exceptions that can be solved more efficiently.

Contribution

It establishes tight bounds for quantum query complexity of minor-closed graph properties and introduces quantum walk algorithms for subgraph detection.

Findings

Most minor-closed properties have quantum query complexity (n^{3/2})

Properties characterized by finitely many forbidden subgraphs can be solved faster

Quantum walk algorithms improve bounds for subgraph-finding problems

Abstract

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.