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:1903.07493 · ACM Symposium on Theory of Computing (STOC)

Quadratic speedup for finding marked vertices by Quantum walks

Andris Ambainis, András Gilyén, Stacey Jeffery, Martins Kokainis

Published Jun 7, 2020• 45 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1145/3357713.3384252

Related papers

Fixed-point Quantum Walk Search.

Lvzhou Li, Yongzhen Xu, Zhang Delong

n this paper, we consider the problem of searching a marked vertex in a graph based on quantum walks. A very large body of research exists on this issue, from both theoretical and experimental studies, yet the existing quantum walk-based search algorithms require knowing exactly how many marked vertices there are before starting the search process. In addition, even if the precise knowledge of the number of marked vertices is known, the success probability of these algorithms could shrink dramatically when the number of search steps is greater than the right one. For avoiding these defects, in this paper we present a new quantum walk-based search algorithm which need not know any prior information about the marked vertices, but still keeps a quadratic speedup over classical ones and ensures that the error is bounded by a tunable parameter $\epsilon$. More specifically, for an $N$-vertex complete bipartite graph with marked vertices but without knowing the number of marked vertices, if the number of search steps $h$ satisfies $h \geq \ln(\frac{2}{\sqrt{\epsilon}})\sqrt{N}$, then the algorithm will output a marked vertex with probability at least $ 1-\epsilon$ for any given $\epsilon\in (0,1]$. This feature leads to quantum search algorithms with stronger robustness and a wider scope of application.

•Nov 17, 2021•0 citations

Quadratic Speedup for Spatial Search by Continuous-Time Quantum Walk

Simon Apers, Shantanav Chakraborty, Leonardo Novo +1 more

Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this Letter, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analogue procedure to perform operations on a state of the form e^{-tH^{2}}|ψ⟩, which, for a given Hamiltonian H, only requires evolving H for time scaling as sqrt[t]. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our Letter thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.

arXiv:2112.12746•Physical Review Letters•Oct 14, 2022•65 citations

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

Quantum walk on two-dimensional torus and grid

Mojtaba Komeili

Let χ be a function that maps the vertices of a graph to zero or one. We want to find a vertex m, called a marked vertex, such that χ(m) = 1. Random walk is an algorithm for finding such a vertex. The measure of complexity for random walk is hitting time. There is a quantum algorithm that finds a marked vertex, and runs in the square root of the extended hitting time. For some configurations of marked vertices on certain graphs, the extended hitting time is asymptotically larger than the hitting time. This might cause the quantum algorithm for finding to be less efficient than a random walk. Two-dimensional torus and grid are canonical examples of such graphs. This thesis is about efficient quantum algorithms for finding a marked vertex on a two-dimensional torus. We prove the convexity of the extended hitting time, and provide a new upper bound on it. We show that, with a probability that is more than a constant, random walk on a torus is localized, and it is restricted to a sub-graph of the torus. From these observations, we introduce two algorithms that find a marked vertex on a torus. The complexity of these algorithms is in the order of the square root of the hitting time, up to logarithmic factors. They both start by finding an estimate to the hitting time. Using that, they estimate the size of a sub-graph that contains a marked vertex with more than a constant probability, and sample such a sub-graph. Our first algorithm finds a marked vertex on this sub-graph by using a quantum algorithm that runs in extended hitting time, and finds a marked vertex with probability at least a constant. Our second algorithm recursively runs the detection algorithm on cuts of the sub-graph, until reaching a constant sized sub-graph that contains a marked vertex. These algorithms are the first quantum algorithms that achieve close-to-quadratic speed-up in quantum walk on a two-dimensional torus, for all configurations of marked vertices.

•May 28, 2018•1 citation

Quantum-walk-based search algorithms with multiple marked vertices

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

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.

arXiv:2103.12878•Physical Review A•Jun 1, 2021•22 citations
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.
n this paper, we consider the problem of searching a marked vertex in a graph based on quantum walks. A very large body of research exists on this issue, from both theoretical and experimental studies, yet the existing quantum walk-based search algorithms require knowing exactly how many marked vertices there are before starting the search process. In addition, even if the precise knowledge of the number of marked vertices is known, the success probability of these algorithms could shrink dramatically when the number of search steps is greater than the right one. For avoiding these defects, in this paper we present a new quantum walk-based search algorithm which need not know any prior information about the marked vertices, but still keeps a quadratic speedup over classical ones and ensures that the error is bounded by a tunable parameter ϵ\epsilonϵ. More specifically, for an NNN-vertex complete bipartite graph with marked vertices but without knowing the number of marked vertices, if the number of search steps hhh satisfies h≥ln⁡(2ϵ)Nh \geq \ln(\frac{2}{\sqrt{\epsilon}})\sqrt{N}h≥ln(ϵ​2​)N​, then the algorithm will output a marked vertex with probability at least $ 1-\epsilon$ for any given ϵ∈(0,1]\epsilon\in (0,1]ϵ∈(0,1]. This feature leads to quantum search algorithms with stronger robustness and a wider scope of application.
Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this Letter, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analogue procedure to perform operations on a state of the form e^{-tH^{2}}|ψ⟩, which, for a given Hamiltonian H, only requires evolving H for time scaling as sqrt[t]. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our Letter thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.
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.
Let χ be a function that maps the vertices of a graph to zero or one. We want to find a vertex m, called a marked vertex, such that χ(m) = 1. Random walk is an algorithm for finding such a vertex. The measure of complexity for random walk is hitting time. There is a quantum algorithm that finds a marked vertex, and runs in the square root of the extended hitting time. For some configurations of marked vertices on certain graphs, the extended hitting time is asymptotically larger than the hitting time. This might cause the quantum algorithm for finding to be less efficient than a random walk. Two-dimensional torus and grid are canonical examples of such graphs. This thesis is about efficient quantum algorithms for finding a marked vertex on a two-dimensional torus. We prove the convexity of the extended hitting time, and provide a new upper bound on it. We show that, with a probability that is more than a constant, random walk on a torus is localized, and it is restricted to a sub-graph of the torus. From these observations, we introduce two algorithms that find a marked vertex on a torus. The complexity of these algorithms is in the order of the square root of the hitting time, up to logarithmic factors. They both start by finding an estimate to the hitting time. Using that, they estimate the size of a sub-graph that contains a marked vertex with more than a constant probability, and sample such a sub-graph. Our first algorithm finds a marked vertex on this sub-graph by using a quantum algorithm that runs in extended hitting time, and finds a marked vertex with probability at least a constant. Our second algorithm recursively runs the detection algorithm on cuts of the sub-graph, until reaching a constant sized sub-graph that contains a marked vertex. These algorithms are the first quantum algorithms that achieve close-to-quadratic speed-up in quantum walk on a two-dimensional torus, for all configurations of marked vertices.
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.