Quantum Query Complexity of Hypergraph Search Problems

Abstract

In the study of quantum query complexity, it is natural to study the problems of finding triangles and spanning trees in a simple graph. Over the past decades, many techniques are developed for finding the upper and lower quantum query bounds of these graph problems. We can generalize these problems to detecting certain properties of higher rank hypergraphs and ask whether these techniques are still available. In this thesis, we will see that when the rank increase, complexity bounds still holds for some problems, although less effectively. For some other problems, their nontrivial complexity bounds vanish. Moreover, we will focused on using the generalized adversary and learning graph techniques for finding nontrivial quantum query bounds for different hypergraph search problems. The following results are presented.

• Discover a general quantum query lower bound for subhypergraph-closed properties and monotone properties over r-partite r-uniform hypergraphs. • Provide tight quantum query bounds for the connectivity and acyclicity problems over r-uniform hypergraphs. • Present a nontrivial learning graph algorithm for the 3-simplex finding problem. • Formulate nested quantum walk in the adaptive learning context and use it to present a nontrivial quantum query algorithm for the 4-simplex finding problem. • Present a natural relationship of lower bounds for simplex finding of different ranks. • Use the learning graph formalization of tetrahedron certificate structure to find a nontrivial quantum query lower bound of the 3-simplex sum problem.