Quantum property testing in sparse directed graphs

2024-10-07

arXiv.org

AI Summary

This work investigates the potential for quantum speedup in property testing on sparse directed graphs, focusing on the one-way query model (where only outgoing edges of vertices can be queried). For the problems of *k*-star-freeness and, more generally, *k*-source-subgraph-freeness, the paper establishes the first theoretical framework for quantum property testing on sparse directed graphs. Using quantum query complexity analysis, dual polynomial techniques, and a novel variant of the *k*-collision problem, it demonstrates a near-quadratic quantum speedup—achieving query complexity $widetilde{O}(sqrt{n})$—and proves this upper bound is nearly optimal, as the classical query complexity is $Omega(n)$. Conversely, the paper reveals fundamental limitations of quantum acceleration: in the bounded-degree undirected graph model, testing 3-colorability still requires $Omega(n)$ quantum queries, precluding any superlinear speedup. Collectively, these results provide a systematic characterization of the boundary of quantum advantage in graph property testing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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 prove 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.

Problem

• Quantum testing of k-source-subgraph-freeness in sparse directed graphs
• Quantum advantage for k-star-freeness testing in unidirectional models
• Quantum limitations in testing 3-colorability for undirected bounded-degree graphs

Innovation

• Quantum testing for sparse directed graphs
• Quadratic advantage in k-source-subgraph-freeness
• Linear queries needed for 3-colorability testing