Quantum property testing in sparse directed graphs

TL;DR

This paper explores quantum algorithms for property testing in sparse directed graphs, revealing significant quantum speedups for certain problems like $k$-star-freeness, but not for others such as 3-colorability.

Contribution

It introduces quantum property testing in sparse directed graphs, demonstrating near-quadratic quantum speedup for $k$-star-freeness and establishing tight bounds, while also showing limitations for testing 3-colorability.

Findings

Quantum algorithms achieve near-quadratic speedup for $k$-star-freeness.

Quantum lower bounds are established using dual polynomials.

Not all graph properties, like 3-colorability, benefit from quantum speedup.

Abstract

We initiate the study of quantum property testing in sparse directed graphs, and more particularly in the unidirectional model, where the algorithm is allowed to query only the outgoing edges of a vertex. In the classical unidirectional model, the problem of testing $k$-star-freeness, and more generally $k$-source-subgraph-freeness, is almost maximally hard for large $k$. We prove that this problem has almost quadratic advantage in the quantum setting. Moreover, we show that this advantage is nearly tight, by showing a quantum lower bound using the method of dual polynomials on an intermediate problem for a new, property testing version of the $k$-collision problem that was not studied before. To illustrate that not all problems in graph property testing admit such a quantum speedup, we consider the problem of $3$-colorability in the related undirected bounded-degree model, when graphs are now undirected. This problem is maximally hard to test classically, and we show that also quantumly it requires a linear number of queries.