Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0209131 · ACM Symposium on Theory of Computing (STOC)

Exponential algorithmic speedup by a quantum walk

Andrew M. Childs, Richard Cleve, E. Deotto, Edward Farhi, Sam Gutmann, Daniel Alan Spielman

Published Jun 9, 2003• 822 citations
arXivHTMLPDF

Abstract

We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.

DOI
10.1145/780542.780552

Related papers

Exponential algorithmic speedup by a quantum walk

Andrew M. Childs, Richard Cleve, E. Deotto +3 more

We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.

ACM Symposium on Theory of Computing (STOC)•Jan 1, 2003•134 citations

A note on the classical lower bound for a quantum walk algorithm

Stephen Fenner, Zhang, Yong

A recent paper on quantum walks by Childs et al. [STOC'03] provides an example of a black-box problem for which there is a quantum algorithm with exponential speedup over the best classical randomized algorithm for the problem, but where the quantum algorithm does not involve any use of the quantum Fourier transform. They give an exponential lower bound for a classical randomized algorithm solving the black-box graph traversal problem defined in their paper. In this note we give an improved lower bound for this problem via a straightforward and more complete analysis.

arXiv:quant-ph/0312230•Dec 31, 2003•9 citations

Quantum Walk-Based Algorithms for Graph Traversal

Jincheng Zhang

This paper investigates the application of quantum walk algorithms to graph traversal problems. Traditional graph traversal algorithms often suffer from performance limitations when dealing with large and complex graphs due to their inherent sequential nature. We propose utilizing quantum walks, quantum analogs of random walks, as a potential solution. Quantum walks leverage quantum superposition and interference to enable more efficient exploration of the graph. We present a theoretical framework for designing quantum walk-based algorithms for graph traversal, focusing on key aspects such as Hamiltonian construction, walker evolution, and measurement strategies. Our work demonstrates the potential for exponential speedups for specific graph algorithms compared to their classical counterparts. We analyze the core principles underpinning this approach and highlight the advantages of quantum walk based algorithms for graph traversal.

•Sep 5, 2026•0 citations

Quantum Walk-Based Algorithms for Graph Traversal and Search

Jincheng Zhang

This paper explores the application of quantum walk algorithms to graph traversal and search problems. Classical algorithms often struggle with the computational complexity of traversing and searching large graphs, leading to significant time and resource limitations. Quantum walks, utilizing superposition and interference, offer a potential pathway to achieve exponential speedups. This work details the theoretical framework for applying quantum walks to graph problems, specifically focusing on traversal and search tasks. We present a generalized approach for mapping graph structures onto quantum walk systems and discuss the key mathematical formulations involved. The core of our investigation revolves around the construction of quantum walk operators that mimic the movement and interaction rules inherent in graph traversal. We analyze the evolution of the quantum walk state and derive expressions for the probability of reaching a specific node in the graph. Furthermore, we examine the impact of graph topology on the performance of the quantum walk, highlighting the potential for tailored algorithms to exploit specific graph structures. The results demonstrate the theoretical feasibility of quantum walk algorithms for significantly accelerating graph traversal and search operations, contributing to advancements in areas such as network analysis, database management, and artificial intelligence.

•Sep 6, 2026•0 citations

Quantum Walk-Based Graph Search Algorithms

Jincheng Zhang

This paper explores the application of quantum walk algorithms for graph search problems. Classical graph search algorithms, such as Dijkstra's algorithm, frequently encounter limitations when dealing with large graphs, often exhibiting performance bottlenecks due to exhaustive exploration of the graph's nodes. We propose a novel approach utilizing quantum walks to address these challenges. Quantum walks exploit quantum superposition and interference to facilitate a more efficient exploration of the graph space compared to their classical counterparts. The theoretical framework presented investigates the potential for exponential speedups in graph search operations. We formally define the quantum walk model and its implementation for graph traversal, outlining the key steps involved in constructing a quantum algorithm. This work focuses on the theoretical aspects of the algorithm, providing a foundation for future experimental investigations and optimization.

•Sep 5, 2026•0 citations

Quantum Walk-Based Algorithms for Large-Scale Graph Search

Jincheng Zhang

This paper investigates the application of quantum walk algorithms to large-scale graph search problems. Classical algorithms often struggle with the computational complexity of exploring vast graphs, particularly when searching for specific patterns or nodes. Quantum walks, inspired by the probabilistic nature of random walks, provide a promising alternative by utilizing quantum superposition and interference to accelerate the search process. We explore the theoretical framework and potential advantages of employing quantum walks for graph traversal, focusing on scenarios where exponential speedups are theoretically possible. This work aims to demonstrate the potential of quantum computing to tackle computationally demanding graph search tasks that are currently intractable for classical methods. The core concept leverages the ability to simultaneously explore multiple paths within a graph, drastically reducing the search time compared to sequential exploration. We present a foundational understanding of quantum walk implementation and analyze its suitability for various graph search algorithms.

•Aug 28, 2026•0 citations
We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.
We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.
A recent paper on quantum walks by Childs et al. [STOC'03] provides an example of a black-box problem for which there is a quantum algorithm with exponential speedup over the best classical randomized algorithm for the problem, but where the quantum algorithm does not involve any use of the quantum Fourier transform. They give an exponential lower bound for a classical randomized algorithm solving the black-box graph traversal problem defined in their paper. In this note we give an improved lower bound for this problem via a straightforward and more complete analysis.
This paper investigates the application of quantum walk algorithms to graph traversal problems. Traditional graph traversal algorithms often suffer from performance limitations when dealing with large and complex graphs due to their inherent sequential nature. We propose utilizing quantum walks, quantum analogs of random walks, as a potential solution. Quantum walks leverage quantum superposition and interference to enable more efficient exploration of the graph. We present a theoretical framework for designing quantum walk-based algorithms for graph traversal, focusing on key aspects such as Hamiltonian construction, walker evolution, and measurement strategies. Our work demonstrates the potential for exponential speedups for specific graph algorithms compared to their classical counterparts. We analyze the core principles underpinning this approach and highlight the advantages of quantum walk based algorithms for graph traversal.
This paper explores the application of quantum walk algorithms to graph traversal and search problems. Classical algorithms often struggle with the computational complexity of traversing and searching large graphs, leading to significant time and resource limitations. Quantum walks, utilizing superposition and interference, offer a potential pathway to achieve exponential speedups. This work details the theoretical framework for applying quantum walks to graph problems, specifically focusing on traversal and search tasks. We present a generalized approach for mapping graph structures onto quantum walk systems and discuss the key mathematical formulations involved. The core of our investigation revolves around the construction of quantum walk operators that mimic the movement and interaction rules inherent in graph traversal. We analyze the evolution of the quantum walk state and derive expressions for the probability of reaching a specific node in the graph. Furthermore, we examine the impact of graph topology on the performance of the quantum walk, highlighting the potential for tailored algorithms to exploit specific graph structures. The results demonstrate the theoretical feasibility of quantum walk algorithms for significantly accelerating graph traversal and search operations, contributing to advancements in areas such as network analysis, database management, and artificial intelligence.
This paper explores the application of quantum walk algorithms for graph search problems. Classical graph search algorithms, such as Dijkstra's algorithm, frequently encounter limitations when dealing with large graphs, often exhibiting performance bottlenecks due to exhaustive exploration of the graph's nodes. We propose a novel approach utilizing quantum walks to address these challenges. Quantum walks exploit quantum superposition and interference to facilitate a more efficient exploration of the graph space compared to their classical counterparts. The theoretical framework presented investigates the potential for exponential speedups in graph search operations. We formally define the quantum walk model and its implementation for graph traversal, outlining the key steps involved in constructing a quantum algorithm. This work focuses on the theoretical aspects of the algorithm, providing a foundation for future experimental investigations and optimization.
This paper investigates the application of quantum walk algorithms to large-scale graph search problems. Classical algorithms often struggle with the computational complexity of exploring vast graphs, particularly when searching for specific patterns or nodes. Quantum walks, inspired by the probabilistic nature of random walks, provide a promising alternative by utilizing quantum superposition and interference to accelerate the search process. We explore the theoretical framework and potential advantages of employing quantum walks for graph traversal, focusing on scenarios where exponential speedups are theoretically possible. This work aims to demonstrate the potential of quantum computing to tackle computationally demanding graph search tasks that are currently intractable for classical methods. The core concept leverages the ability to simultaneously explore multiple paths within a graph, drastically reducing the search time compared to sequential exploration. We present a foundational understanding of quantum walk implementation and analyze its suitability for various graph search algorithms.