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

2025-08-16

AI Summary

This work reevaluates the asymptotic quantum advantage for the single-source shortest paths (SSSP) problem in light of recent advances in classical algorithms—specifically, the improved classical algorithm by Duan et al. Method: We conduct a systematic theoretical complexity comparison among Dijkstra’s algorithm, Duan’s classical algorithm, and quantum algorithms by Wesolowski–Piddock et al., across varying graph densities and shortest-path lengths $L$. We introduce a dynamic framework for assessing quantum advantage conditioned on structural parameters. Results: We establish that quantum algorithms achieve strict asymptotic advantage *only* when the shortest-path length satisfies $L = o(n^{1/3})$; for larger $L$, state-of-the-art classical algorithms consistently dominate. This demonstrates that progress in classical algorithm design is concretely narrowing the regime of quantum supremacy for SSSP. Moreover, our analysis provides a new paradigm for designing practical quantum algorithms tailored to real-world constraints—including path-scale limitations and graph topology—thereby shifting focus from unconditional speedups to context-aware quantum advantage.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

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.

Problem

• Compare quantum and classical shortest path algorithms' performance.
• Analyze scaling in different graph density and path length scenarios.
• Re-evaluate quantum advantage with new classical algorithm benchmarks.

Innovation

• Compares quantum and classical shortest path algorithms
• Analyzes asymptotic scaling in varied graph scenarios
• Highlights quantum advantage in short path regimes