A quantum algorithm for learning a graph of bounded degree

• URL: http://arxiv.org/abs/2402.18714v1
• Date: Wed, 28 Feb 2024 21:23:40 GMT
• Title: A quantum algorithm for learning a graph of bounded degree
• Authors: Asaf Ferber, Liam Hardiman
• Abstract summary: We present an algorithm that learns the edges of $G$ in at most $tildeO(d2m3/4)$ quantum queries. In particular, we present a randomized algorithm that, with high probability, learns cycles and matchings in $tildeO(sqrtm)$ quantum queries.
• Score: 1.8130068086063336
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We are presented with a graph, $G$, on $n$ vertices with $m$ edges whose edge set is unknown. Our goal is to learn the edges of $G$ with as few queries to an oracle as possible. When we submit a set $S$ of vertices to the oracle, it tells us whether or not $S$ induces at least one edge in $G$. This so-called OR-query model has been well studied, with Angluin and Chen giving an upper bound on the number of queries needed of $O(m \log n)$ for a general graph $G$ with $m$ edges. When we allow ourselves to make *quantum* queries (we may query subsets in superposition), then we can achieve speedups over the best possible classical algorithms. In the case where $G$ has maximum degree $d$ and is $O(1)$-colorable, Montanaro and Shao presented an algorithm that learns the edges of $G$ in at most $\tilde{O}(d^2m^{3/4})$ quantum queries. This gives an upper bound of $\tilde{O}(m^{3/4})$ quantum queries when $G$ is a matching or a Hamiltonian cycle, which is far away from the lower bound of $\Omega(\sqrt{m})$ queries given by Ambainis and Montanaro. We improve on the work of Montanaro and Shao in the case where $G$ has bounded degree. In particular, we present a randomized algorithm that, with high probability, learns cycles and matchings in $\tilde{O}(\sqrt{m})$ quantum queries, matching the theoretical lower bound up to logarithmic factors.

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