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:2103.12878 · Physical Review A

Quantum-walk-based search algorithms with multiple marked vertices

Gustavo Alves Bezerra, Pedro H. G. Lugão, Renato Portugal

Published Jun 1, 2021• 22 citations
arXivHTMLPDF

Abstract

The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, with Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum-walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.

DOI
10.1103/physreva.103.062202

Related papers

Quantum Walk Search on Complete Multipartite Graph with Multiple Marked Vertices

Nan-Yow Chen, Meng Li, Xiaoming Sun

Quantum walk is a potent technique for building quantum algorithms. This paper examines the quantum walk search algorithm on complete multipartite graphs with multiple marked vertices, which has not been explored before. Two specific cases of complete multipartite graphs are probed in this paper, and in both cases, each set consists of an equal number of vertices. We employ the coined quantum walk model and achieve quadratic speedup with a constant probability of finding a marked vertex. Furthermore, we investigate the robust quantum walk of two cases and demonstrate that even with an unknown number of marked vertices, it is still possible to achieve a quadratic speedup compared to classical algorithms and the success probability oscillates within a small range close to 1. This work addresses the overcooking problem in quantum walk search algorithms on some complete multipartite graphs. We also provide the numerical simulation and circuit implementation of our quantum algorithm.

arXiv:2410.04924•Oct 7, 2024•0 citations

Robust Quantum Walk Search on Complete Multipartite Graph With Multiple Marked Vertices

Nan-Yow Chen, Meng Li, Xiaoming Sun

Quantum walks are a potent technique for building quantum algorithms. This paper examines the quantum walk search algorithm on complete multipartite graphs with multiple marked vertices, which has not been explored before. We employ the coined quantum walk model and achieve quadratic speedup with a constant probability of finding a marked vertex in two specific cases of complete multipartite graphs. Furthermore, we investigate the robust quantum walk of two cases and demonstrate that even with an unknown number of marked vertices, it is still possible to achieve a quadratic speedup compared to classical algorithms and the success probability oscillates within a small range close to 1. To establish this conclusion, we employ approximation and reduction methods to conduct theoretical analysis and proofs, and provide an intuitive demonstration through numerical simulations. This work addresses the overcooking problem in quantum walk search algorithms on some complete multipartite graphs and may offer insights for robust quantum search on broader graph structures. Finally, we design a circuit for the algorithm that future researchers can use to conduct experiments.

IEEE Transactions on Quantum Engineering•Jan 1, 2026•1 citation

Multimarked Spatial Search by Continuous-Time Quantum Walk

Pedro H. G. Lugão, Renato Portugal, Mohamed Sabri +1 more

The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian derived from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained from the roots of the determinant of a real symmetric matrix M , the dimensions of which depend on the number of marked vertices. The eigenvectors are determined from a basis of the kernel of M . We show each step of the framework by solving the spatial searching problem on the Johnson graphs with a fixed diameter and with two marked vertices. Our calculations show that the optimal running time is \(O(\sqrt {N})\) with an asymptotic probability of 1+ o (1), where N is the number of vertices.

arXiv:2203.14384•ACM Transactions on Quantum Computing•Nov 27, 2024•5 citations

Adjacent vertices can be hard to find by quantum walks

Nikolajs Nahimovs, Raqueline A. M. Santos, Kamil Ravilevich Khadiev

Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs Ω(N) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph. Additionally, we consider an algorithmic application of the found effect. We investigate a problem of detection of a perfect matching in a bipartite graph. We use the found effect as an algorithmic building block and construct quantum algorithm which, for a specific class of graphs, outperforms its classical analogs.

•May 14, 2018•1 citation

Adjacent Vertices Can be Hard to Find by Quantum Walks

Nikolajs Nahimovs, Raqueline A. M. Santos, K. R. Khadiev

Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs Ω( N ) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph. Additionally, we consider an algorithmic application of the found effect. We investigate a problem of detection of a perfect matching in a bipartite graph. We use the found effect as an algorithmic building block and construct quantum algorithm which, for a specific class of graphs, outperforms its classical analogs.

arXiv:1605.05598•Moscow University Computational Mathematics and Cybernetics•Jan 1, 2019•6 citations

Quadratic speedup for finding marked vertices by Quantum walks

Andris Ambainis, András Gilyén, Stacey Jeffery +1 more

A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk, resolving a question that had been open for 15 years.

arXiv:1903.07493•ACM Symposium on Theory of Computing (STOC)•Jun 7, 2020•45 citations
The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, with Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum-walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.
Quantum walk is a potent technique for building quantum algorithms. This paper examines the quantum walk search algorithm on complete multipartite graphs with multiple marked vertices, which has not been explored before. Two specific cases of complete multipartite graphs are probed in this paper, and in both cases, each set consists of an equal number of vertices. We employ the coined quantum walk model and achieve quadratic speedup with a constant probability of finding a marked vertex. Furthermore, we investigate the robust quantum walk of two cases and demonstrate that even with an unknown number of marked vertices, it is still possible to achieve a quadratic speedup compared to classical algorithms and the success probability oscillates within a small range close to 1. This work addresses the overcooking problem in quantum walk search algorithms on some complete multipartite graphs. We also provide the numerical simulation and circuit implementation of our quantum algorithm.
Quantum walks are a potent technique for building quantum algorithms. This paper examines the quantum walk search algorithm on complete multipartite graphs with multiple marked vertices, which has not been explored before. We employ the coined quantum walk model and achieve quadratic speedup with a constant probability of finding a marked vertex in two specific cases of complete multipartite graphs. Furthermore, we investigate the robust quantum walk of two cases and demonstrate that even with an unknown number of marked vertices, it is still possible to achieve a quadratic speedup compared to classical algorithms and the success probability oscillates within a small range close to 1. To establish this conclusion, we employ approximation and reduction methods to conduct theoretical analysis and proofs, and provide an intuitive demonstration through numerical simulations. This work addresses the overcooking problem in quantum walk search algorithms on some complete multipartite graphs and may offer insights for robust quantum search on broader graph structures. Finally, we design a circuit for the algorithm that future researchers can use to conduct experiments.
The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian derived from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained from the roots of the determinant of a real symmetric matrix M , the dimensions of which depend on the number of marked vertices. The eigenvectors are determined from a basis of the kernel of M . We show each step of the framework by solving the spatial searching problem on the Johnson graphs with a fixed diameter and with two marked vertices. Our calculations show that the optimal running time is O(N)O(\sqrt {N})O(N​) with an asymptotic probability of 1+ o (1), where N is the number of vertices.
Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs Ω(N) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph. Additionally, we consider an algorithmic application of the found effect. We investigate a problem of detection of a perfect matching in a bipartite graph. We use the found effect as an algorithmic building block and construct quantum algorithm which, for a specific class of graphs, outperforms its classical analogs.
Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs Ω( N ) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph. Additionally, we consider an algorithmic application of the found effect. We investigate a problem of detection of a perfect matching in a bipartite graph. We use the found effect as an algorithmic building block and construct quantum algorithm which, for a specific class of graphs, outperforms its classical analogs.
A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk, resolving a question that had been open for 15 years.