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:2104.03806 · Physical Review A

Deterministic spatial search using alternating quantum walks

S. Marsh, Jingbo B. Wang

Published Aug 24, 2021• 15 citations
arXivHTMLPDF

Abstract

This paper examines the performance of spatial search where the Grover diffusion operator is replaced by continuous-time quantum walks on a class of interdependent networks. We prove that, for a set of optimal quantum walk times and marked vertex phase shifts, a deterministic algorithm for structured spatial search is established that finds the marked vertex with 100% probability. This improves on the Childs and Goldstone spatial search algorithm on the same class of graphs, which we show can only amplify to 50% probability. Our method uses $\ensuremath{\lceil}\frac{\ensuremath{\pi}}{2\sqrt{2}}\sqrt{N}\ensuremath{\rceil}$ marked vertex phase shifts for an $N$-vertex graph, making it comparable to Grover's algorithm for an unstructured search. It is expected that this framework can be readily extended to deterministic spatial search on other families of graph structures.

DOI
10.1103/physreva.104.022216

Related papers

Universal approach to deterministic spatial search via alternating quantum walks

Qing‐Wen Wang, Ying Jiang, Shiguang Feng +1 more

Spatial search is an important problem in quantum computation, which aims to find a marked vertex on a graph. We propose a novel approach for designing deterministic quantum search algorithms on a variety of graphs via alternating quantum walks. Our approach is universal because it does not require an instance-specific analysis for different graphs. We highlight the flexibility of our approach by proving that for Johnson graphs, rook graphs, complete-square graphs and complete bipartite graphs, our quantum algorithms can find the marked vertex with $100\%$ success probability and achieve quadratic speedups over classical algorithms. This not only gives an alternative succinct way to prove the existing results, but also leads to new interesting findings on more general graphs.

arXiv:2307.16133•Jul 30, 2023•2 citations

A framework for optimal quantum spatial search using alternating phase-walks

S. Marsh, Jingbo B. Wang

Abstract We present a novel methodological framework for quantum spatial search, generalising the Childs & Goldstone ( CG ) algorithm via alternating applications of marked-vertex phase shifts and continuous-time quantum walks. We determine closed form expressions for the optimal walk time and phase shift parameters for periodic graphs. These parameters are chosen to rotate the system about subsets of the graph Laplacian eigenstates, amplifying the probability of measuring the marked vertex. The state evolution is asymptotically optimal for any class of periodic graphs having a fixed number of unique eigenvalues. We demonstrate the effectiveness of the algorithm by applying it to obtain O ( N ) search on a variety of graphs. One important class is the n × n 3 rook graph, which has N = n 4 vertices. On this class of graphs the CG algorithm performs suboptimally, achieving only O ( N − 1 / 8 ) overlap after time O ( N 5 / 8 ) . Using the new alternating phase-walk framework, we show that O ( 1 ) overlap is obtained in O ( N ) phase-walk iterations.

arXiv:2109.14351•Quantum Science and Technology•Sep 14, 2021•4 citations

Deterministic Quantum Search on All Laplacian Integral Graphs

Guanzhong Li, Jingquan Luo, Shiguang Feng +1 more

Abstract Searching for an unknown marked vertex on a given graph (also known as spatial search) is an extensively discussed topic in the area of quantum algorithms, with a plethora of results based on different quantum walk models and targeting various types of graphs. Most of these algorithms have a non‐zero probability of failure. In recent years, there have been some efforts to design quantum spatial search algorithms with success probability. However, these works either only work for very special graphs or only for the case where there is only one marked vertex. In this work, a different and elegant approach to quantum spatial search is proposed, obtaining deterministic quantum search algorithms that can find a marked vertex with certainty on any Laplacian integral graph with any predetermined proportion of marked vertices. Thus, this work discovers the largest class of graphs that allow deterministic quantum search, making it easy to design deterministic quantum search algorithms for many graphs, subsuming different graphs discussed in previous works, in a unified framework.

arXiv:2506.21108•Advanced Quantum Technologies•Oct 30, 2025•2 citations

Deterministic discrete-time quantum walk search on complete bipartite graphs

Fangjie Peng, Meng Li, Xiaoming Sun

Searching via quantum walk is a topic that has been extensively studied. Most previous results provide approximate solutions, while in this paper we prove an algorithm that can find a marked vertex certainly. We adopt the coined discrete-time quantum walk (DTQW) model with adjusted operators and prove that, on complete bipartite graphs, when parameters are set properly, coined DTQW can deterministically find a marked vertex, i.e., the success probability is exactly 1. Before this paper there have been results of an alternating continuous-time quantum walk method that achieve deterministic spatial search, but this paper provides a deterministic quantum spatial search result via DTQW, while maintaining a quadratic speedup compared to classical algorithms. We also provide the quantum circuit implementation. Published by the American Physical Society 2024

Physical Review Research•Jul 9, 2024•5 citations

Unifying quantum spatial search, state transfer, and uniform sampling on graphs

Qingwen Wang, Ying Jiang, Shiguang Feng +1 more

In this article, we present a succinct algorithmic framework via alternating quantum walks, unifying quantum spatial search, state transfer, and uniform sampling on a large class of graphs. Using the framework, we can achieve exact uniform sampling over all vertices and perfect state transfer between any two vertices, provided that eigenvalues of Laplacian matrix of the graph are all integers. Furthermore, if the graph is vertex transitive as well, then we can achieve deterministic quantum spatial search that finds a marked vertex with probability $100%$. In contrast, existing quantum search algorithms generally have a certain probability of failure. Even if the graph is not vertex transitive, such as the complete bipartite graph, we can still adjust the algorithmic framework to obtain deterministic spatial search, which thus shows the flexibility of it. Besides unifying and improving plenty of previous results, our work provides results on more graphs. The approach is easy to use since it has a succinct formalism that depends only on the depth of the Laplacian eigenvalue set of the graph, and may shed light on the solution of more problems related to graphs.

arXiv:2407.02530•Physical Review A•Apr 9, 2025•6 citations

Deterministic Search on Complete Bipartite Graphs by Continuous-Time Quantum Walk

Honghong Lin, Yun Shang

Abstract How to design deterministic spatial search algorithms is a difficult but important problem. This paper presents a deterministic search algorithm on complete bipartite graphs with optimal runtime scaling. Our algorithm adopts the simple form of alternating iterations of an oracle and a continuous-time quantum walk operator, which is a generalization of Grover's search algorithm. To address the case of the number of marked vertices being unknown, we construct a quantum counting algorithm based on the spectrum structure of the search operator. This is a non-trivial example of quantum counting for spatial search. To implement the continuous-time quantum walk operator, we perform Hamiltonian simulation in the quantum circuit model and simulate it on the IBM quantum computing platform Qiskit.

arXiv:2404.01640•Quantum Information and Computation•Oct 1, 2024•2 citations
This paper examines the performance of spatial search where the Grover diffusion operator is replaced by continuous-time quantum walks on a class of interdependent networks. We prove that, for a set of optimal quantum walk times and marked vertex phase shifts, a deterministic algorithm for structured spatial search is established that finds the marked vertex with 100% probability. This improves on the Childs and Goldstone spatial search algorithm on the same class of graphs, which we show can only amplify to 50% probability. Our method uses \ensuremath⌈\ensuremathπ22N\ensuremath⌉\ensuremath{\lceil}\frac{\ensuremath{\pi}}{2\sqrt{2}}\sqrt{N}\ensuremath{\rceil}\ensuremath⌈22​\ensuremathπ​N​\ensuremath⌉ marked vertex phase shifts for an NNN-vertex graph, making it comparable to Grover's algorithm for an unstructured search. It is expected that this framework can be readily extended to deterministic spatial search on other families of graph structures.
Spatial search is an important problem in quantum computation, which aims to find a marked vertex on a graph. We propose a novel approach for designing deterministic quantum search algorithms on a variety of graphs via alternating quantum walks. Our approach is universal because it does not require an instance-specific analysis for different graphs. We highlight the flexibility of our approach by proving that for Johnson graphs, rook graphs, complete-square graphs and complete bipartite graphs, our quantum algorithms can find the marked vertex with 100%100\%100% success probability and achieve quadratic speedups over classical algorithms. This not only gives an alternative succinct way to prove the existing results, but also leads to new interesting findings on more general graphs.
Abstract We present a novel methodological framework for quantum spatial search, generalising the Childs & Goldstone ( CG ) algorithm via alternating applications of marked-vertex phase shifts and continuous-time quantum walks. We determine closed form expressions for the optimal walk time and phase shift parameters for periodic graphs. These parameters are chosen to rotate the system about subsets of the graph Laplacian eigenstates, amplifying the probability of measuring the marked vertex. The state evolution is asymptotically optimal for any class of periodic graphs having a fixed number of unique eigenvalues. We demonstrate the effectiveness of the algorithm by applying it to obtain O ( N ) search on a variety of graphs. One important class is the n × n 3 rook graph, which has N = n 4 vertices. On this class of graphs the CG algorithm performs suboptimally, achieving only O ( N − 1 / 8 ) overlap after time O ( N 5 / 8 ) . Using the new alternating phase-walk framework, we show that O ( 1 ) overlap is obtained in O ( N ) phase-walk iterations.
Abstract Searching for an unknown marked vertex on a given graph (also known as spatial search) is an extensively discussed topic in the area of quantum algorithms, with a plethora of results based on different quantum walk models and targeting various types of graphs. Most of these algorithms have a non‐zero probability of failure. In recent years, there have been some efforts to design quantum spatial search algorithms with success probability. However, these works either only work for very special graphs or only for the case where there is only one marked vertex. In this work, a different and elegant approach to quantum spatial search is proposed, obtaining deterministic quantum search algorithms that can find a marked vertex with certainty on any Laplacian integral graph with any predetermined proportion of marked vertices. Thus, this work discovers the largest class of graphs that allow deterministic quantum search, making it easy to design deterministic quantum search algorithms for many graphs, subsuming different graphs discussed in previous works, in a unified framework.
Searching via quantum walk is a topic that has been extensively studied. Most previous results provide approximate solutions, while in this paper we prove an algorithm that can find a marked vertex certainly. We adopt the coined discrete-time quantum walk (DTQW) model with adjusted operators and prove that, on complete bipartite graphs, when parameters are set properly, coined DTQW can deterministically find a marked vertex, i.e., the success probability is exactly 1. Before this paper there have been results of an alternating continuous-time quantum walk method that achieve deterministic spatial search, but this paper provides a deterministic quantum spatial search result via DTQW, while maintaining a quadratic speedup compared to classical algorithms. We also provide the quantum circuit implementation. Published by the American Physical Society 2024
In this article, we present a succinct algorithmic framework via alternating quantum walks, unifying quantum spatial search, state transfer, and uniform sampling on a large class of graphs. Using the framework, we can achieve exact uniform sampling over all vertices and perfect state transfer between any two vertices, provided that eigenvalues of Laplacian matrix of the graph are all integers. Furthermore, if the graph is vertex transitive as well, then we can achieve deterministic quantum spatial search that finds a marked vertex with probability 100100%100. In contrast, existing quantum search algorithms generally have a certain probability of failure. Even if the graph is not vertex transitive, such as the complete bipartite graph, we can still adjust the algorithmic framework to obtain deterministic spatial search, which thus shows the flexibility of it. Besides unifying and improving plenty of previous results, our work provides results on more graphs. The approach is easy to use since it has a succinct formalism that depends only on the depth of the Laplacian eigenvalue set of the graph, and may shed light on the solution of more problems related to graphs.
Abstract How to design deterministic spatial search algorithms is a difficult but important problem. This paper presents a deterministic search algorithm on complete bipartite graphs with optimal runtime scaling. Our algorithm adopts the simple form of alternating iterations of an oracle and a continuous-time quantum walk operator, which is a generalization of Grover's search algorithm. To address the case of the number of marked vertices being unknown, we construct a quantum counting algorithm based on the spectrum structure of the search operator. This is a non-trivial example of quantum counting for spatial search. To implement the continuous-time quantum walk operator, we perform Hamiltonian simulation in the quantum circuit model and simulate it on the IBM quantum computing platform Qiskit.