Quantum Algorithms for Lowest Weight Paths and Spanning Trees in Complete Graphs

TL;DR

This paper develops quantum algorithms that improve the efficiency of solving classical graph problems like shortest paths, spanning trees, and diameter calculations, especially in complete and bipartite graphs.

Contribution

It introduces quantum modifications to classical algorithms, enhancing their speed for specific graph problems in complete and bipartite graphs.

Findings

Quantum algorithms outperform classical ones in specific graph problems.

Quantum search reduces complexity in minimum spanning tree and shortest path calculations.

Fast quantum diameter computation for complete graphs.

Abstract

Quantum algorithms for several problems in graph theory are considered. Classical algorithms for finding the lowest weight path between two points in a graph and for finding a minimal weight spanning tree involve searching over some space. Modification of classical algorithms due to Dijkstra and Prim allows quantum search to replace classical search and leads to more efficient algorithms. In the case of highly asymmetric complete bipartite graphs, simply replacing classical search with quantum search leads to a faster quantum algorithm. A fast quantum algorithm for computing the diameter of a complete graph is also given.