Span-program-based quantum algorithms for graph bipartiteness and connectivity

TL;DR

This paper introduces two new quantum algorithms based on span programs for testing bipartiteness and connectivity of graphs, achieving query complexities of O(n√n), advancing quantum graph property testing.

Contribution

The paper develops span-program-based quantum algorithms for bipartiteness and connectivity testing, providing explicit query complexities and demonstrating their effectiveness.

Findings

Quantum algorithms for bipartiteness and connectivity with O(n√n) queries

Span programs enable optimal quantum query complexity for graph properties

New methods simplify the design of quantum algorithms for graph problems

Abstract

Span program is a linear-algebraic model of computation which can be used to design quantum algorithms. For any Boolean function there exists a span program that leads to a quantum algorithm with optimal quantum query complexity. In general, finding such span programs is not an easy task. In this work, given a query access to the adjacency matrix of a simple graph $G$ with $n$ vertices, we provide two new span-program-based quantum algorithms: an algorithm for testing if the graph is bipartite that uses $O(n\sqrt{n})$ quantum queries; an algorithm for testing if the graph is connected that uses $O(n\sqrt{n})$ quantum queries.