Quantum Query Complexity of Subgraph Isomorphism and Homomorphism

Let $H$ be a fixed graph on $n$ vertices. Let $fH(G) = 1$ iff the input graph $G$ on $n$ vertices contains $H$ as a (not necessarily induced) subgraph. Let $\alphaH$ denote the cardinality of a maximum independent set of $H$. In this paper we show: [Q(fH) = \Omega\left(\sqrt{\alphaH \cdot n}\right),] where $Q(fH)$ denotes the quantum query complexity of $fH$. As a consequence we obtain a lower bounds for $Q(fH)$ in terms of several other parameters of $H$ such as the average degree, minimum vertex cover, chromatic number, and the critical probability. We also use the above bound to show that $Q(fH) = \Omega(n^{3/4})$ for any $H$, improving on the previously best known bound of $\Omega(n^{2/3})$. Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our $\Omega(n^{3/4})$ bound for $Q(fH)$ matches the square root of the current best known bound for the randomized query complexity of $fH$, which is $\Omega(n^{3/2})$ due to Gr\"oger. Interestingly, the randomized bound of $\Omega(\alphaH \cdot n)$ for $fH$ still remains open. We also study the Subgraph Homomorphism Problem, denoted by $f{[H]}$, and show that $Q(f{[H]}) = \Omega(n)$. Finally we extend our results to the $3$-uniform hypergraphs. In particular, we show an $\Omega(n^{4/5})$ bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known $\Omega(n^{3/4})$ bound. For the Subgraph Homomorphism, we obtain an $\Omega(n^{3/2})$ bound for the same.