Quantum Query Complexity of Subgraph Containment with Constant-sized Certificates

TL;DR

This paper investigates the quantum query complexity for detecting constant-sized subgraphs within larger graphs, providing improved bounds using the learning graph model, especially for specific subgraphs like triangles and cliques.

Contribution

It introduces a new quantum query complexity bound for subgraph containment problems that improves upon previous results, utilizing the learning graph framework.

Findings

Quantum query complexity for subgraph detection is improved.

The bounds depend on the size of the subgraph and a new function g(H).

Results apply to problems like triangle and clique detection.

Abstract

We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether an $ n $-vertex graph contains a triangle, clique or star of some size. For a general subgraph $ H $ with $ k $ vertices, we show that $ H $ containment can be solved with quantum query complexity $ O(n^{2-\frac{2}{k}-g(H)}) $, with $ g(H) $ a strictly positive function of $ H $. This is better than $ \tilde{O}\s{n^{2-2/k}} $ by Magniez et al. These results are obtained in the learning graph model of Belovs.