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Quantum Computing for Graph Partitioning with Optimal Speedup

Jincheng Zhang

Published Sep 5, 2026• 0 citations
Publisher

Abstract

Graph partitioning is a fundamental problem in computer science with applications in diverse fields like network design, compiler optimization, and data analysis. Traditional algorithms for graph partitioning often suffer from computational complexity, particularly when dealing with large graphs. This paper explores the potential of utilizing quantum computing to accelerate graph partitioning. Specifically, we investigate the application of Grover's algorithm, a quantum search algorithm, to the problem. Our core claim is that by leveraging the superposition and interference capabilities inherent in quantum mechanics, a quantum solution can achieve a provable speedup compared to classical approaches. We present a theoretical framework outlining the algorithm's operation and derive the expected speedup. The key mathematical formulation involves representing the graph partitioning problem as an unstructured search problem amenable to Grover's algorithm. We define the problem as minimizing the cut size, represented by the objective function: *f(S) = Σi∈S Σj∈Sc d(i, j)* where *S* is a subset of vertices, *Sc* is the complement of *S* in the graph, and *d(i, j)* is the shortest path distance between vertices *i* and *j*. The algorithm then constructs a quantum oracle that maps the problem state to a superposition of all possible partitions. Grover's algorithm is then applied to amplify the amplitude of the solution state, resulting in a high probability of finding an optimal or near-optimal partition. The speedup is directly proportional to the square root of the number of iterations performed by Grover's algorithm, which is influenced by the problem size and the quality of the oracle. We analyze the complexity of the quantum algorithm and compare it to the complexity of the best-known classical algorithms, demonstrating a potential for significant speedup. Further research will focus on optimizing the oracle construction and exploring other quantum algorithms suitable for graph partitioning.

DOI
10.5281/zenodo.22318826

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Graph partitioning is a fundamental problem in computer science with applications in diverse fields, including image processing, network analysis, and VLSI design. Traditional graph partitioning algorithms often face significant computational challenges when dealing with large graphs, leading to exponential scaling of execution time. This paper investigates the application of quantum walk algorithms to address this challenge. We propose a framework leveraging quantum walks to explore different graph partitioning configurations, guided by a cost function that quantifies the quality of each partition. The core idea is to use the inherent parallelism of quantum walks to efficiently sample the partition space. The algorithm's performance is evaluated theoretically, aiming to demonstrate a potential speedup compared to classical approaches. This work lays a foundation for further research into quantum graph partitioning algorithms and explores the possibilities of harnessing quantum computing for optimization problems.
This paper investigates the application of quantum walk algorithms for graph partitioning, a fundamental problem in computational graph theory with significant implications for areas like network analysis, circuit design, and software optimization. Classical graph partitioning algorithms often face computational bottlenecks when dealing with large graphs, primarily due to the exhaustive exploration of all possible partitions. We propose a novel approach that utilizes the inherent properties of quantum walks to explore graph connectivity. The core concept involves simulating a quantum walk on the graph, where nodes probabilistically "jump" between adjacent nodes based on defined amplitudes. This approach aims to exploit quantum parallelism and interference to potentially achieve significant speedups compared to traditional methods. The theoretical framework presented outlines the key elements of this algorithm, including the walk's transition probabilities, the measurement process, and the resulting partition. We demonstrate the potential of this method and highlight areas for future research and development.
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Graph partitioning is a fundamental problem in computer science, with applications ranging from high-performance computing to image processing and data analytics. Traditional graph partitioning algorithms often struggle with scalability, particularly when dealing with large, complex graphs. This paper proposes a novel quantum-inspired graph partitioning algorithm designed to address these limitations. The core idea is to mimic the principles of quantum superposition and entanglement to explore the entire solution space simultaneously. The algorithm utilizes probabilistic methods to navigate this space and converge towards a near-optimal partitioning, offering potential improvements in efficiency and scalability compared to conventional methods. The methodology introduces a scoring function based on graph cut metrics and probabilistic routing to simulate quantum interference, ultimately leading to a more effective partitioning. The performance of the algorithm is evaluated through simulations, demonstrating its potential to significantly reduce the computational cost of graph partitioning for large-scale systems. Further research will focus on optimizing the probabilistic routing and exploring different quantum-inspired techniques.