Raising the Bar: An Asymptotic Comparison of Classical and Quantum Shortest Path Algorithms

TL;DR

This paper provides a theoretical comparison of classical and quantum algorithms for the Single-Source Shortest Path problem, revealing that quantum advantages depend on graph properties and are challenged by recent classical improvements.

Contribution

It offers a systematic asymptotic analysis of modern quantum and classical SSSP algorithms considering new classical advancements, updating the quantum advantage landscape.

Findings

Quantum algorithms can outperform classical ones for short paths.

Classical algorithms maintain advantage for long paths.

The relative performance depends on graph density and path length.

Abstract

The Single-Source Shortest Path (SSSP) problem is a cornerstone of computer science with vast applications, for which Dijkstra's algorithm has long been the classical baseline. While various quantum algorithms have been proposed, their performance has typically been benchmarked against this decades-old approach. This landscape was recently reshaped by the introduction of a new classical algorithm by Duan et al. with a complexity of $O(m \cdot (\log n)^{2/3})$. This development necessitates a re-evaluation of the quantum advantage narrative for SSSP. In this paper, we conduct a systematic theoretical comparison of modern quantum and classical SSSP algorithms in light of this new classical frontier. Through an analysis of their theoretical cost functions, we illustrate how their relative scaling compares across scenarios that vary in graph density and path length. Our analysis suggests a nuanced picture: sophisticated quantum algorithms, such as the one by Wesolowski and Piddock, can exhibit more favorable asymptotic scaling, but only in regimes characterized by short solution paths. Conversely, for problems involving long paths, state-of-the-art classical algorithms appear to maintain a scaling advantage. Our work provides an updated perspective for future quantum algorithm development and underscores that the pursuit of quantum advantage is a dynamic race where the classical goalposts are continually shifting.