Time and Space Efficient Quantum Algorithms for Detecting Cycles and Testing Bipartiteness

TL;DR

This paper introduces space and time efficient quantum algorithms for detecting cycles and testing bipartiteness in graphs, achieving significant improvements in runtime and memory usage in different graph models.

Contribution

It presents novel quantum algorithms for graph properties that operate efficiently in both adjacency matrix and adjacency array models, with minimal space requirements.

Findings

Quantum algorithms run in  O(n^{3/2}) time for adjacency matrix model.

Algorithms in adjacency array model run in  O(n \u221a{d_m}) time.

Both algorithms require only O( ) space.

Abstract

We study space and time efficient quantum algorithms for two graph problems -- deciding whether an $n$-vertex graph is a forest, and whether it is bipartite. Via a reduction to the s-t connectivity problem, we describe quantum algorithms for deciding both properties in $\tilde{O}(n^{3/2})$ time and using $O(\log n)$ classical and quantum bits of storage in the adjacency matrix model. We then present quantum algorithms for deciding the two properties in the adjacency array model, which run in time $\tilde{O}(n\sqrt{d_m})$ and also require $O(\log n)$ space, where $d_m$ is the maximum degree of any vertex in the input graph.