Quantum Query Complexity of Subgraph Isomorphism and Homomorphism [article]

Raghav Kulkarni, Supartha Podder
2015 arXiv   pre-print
Let H be a fixed graph on n vertices. Let f_H(G) = 1 iff the input graph G on n vertices contains H as a (not necessarily induced) subgraph. Let α_H denote the cardinality of a maximum independent set of H. In this paper we show: Q(f_H) = Ω(√(α_H · n)), where Q(f_H) denotes the quantum query complexity of f_H. As a consequence we obtain a lower bounds for Q(f_H) in terms of several other parameters of H such as the average degree, minimum vertex cover, chromatic number, and the critical
more » ... ity. We also use the above bound to show that Q(f_H) = Ω(n^3/4) for any H, improving on the previously best known bound of Ω(n^2/3). Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our Ω(n^3/4) bound for Q(f_H) matches the square root of the current best known bound for the randomized query complexity of f_H, which is Ω(n^3/2) due to Gröger. Interestingly, the randomized bound of Ω(α_H · n) for f_H still remains open. We also study the Subgraph Homomorphism Problem, denoted by f_[H], and show that Q(f_[H]) = Ω(n). Finally we extend our results to the 3-uniform hypergraphs. In particular, we show an Ω(n^4/5) bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known Ω(n^3/4) bound. For the Subgraph Homomorphism, we obtain an Ω(n^3/2) bound for the same.
arXiv:1509.06361v2 fatcat:eydgqkabuvhbhdxdxrwarfnqcu