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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Grover’s quantum walk on segmented complete graphs

Gamal Mograby, Benjamin Maloy, Anastasiia Minenkova

Published Oct 9, 2025• 0 citations
Publisher

Abstract

We study Grover's quantum walk on a new class of graphs, termed segmented complete graphs, which combine high symmetry with detailed spectral properties. Using these graphs, we implement Grover's search algorithm and investigate its performance, focussing on the relationship between graph volume, optimal search time and success probabilities. Our results generalize classical findings for directed weighted graphs and provide new insights into enhancing quantum search algorithms on complex graph structures.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.

DOI
10.1098/rsta.2024.0419

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We study Grover's quantum walk on a new class of graphs, termed segmented complete graphs, which combine high symmetry with detailed spectral properties. Using these graphs, we implement Grover's search algorithm and investigate its performance, focussing on the relationship between graph volume, optimal search time and success probabilities. Our results generalize classical findings for directed weighted graphs and provide new insights into enhancing quantum search algorithms on complex graph structures.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.
Abstract We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy’s quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has l integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight l . We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we generalize Grover’s algorithm, or search on the complete graph, to have a weighted self-loop at each vertex, yielding an improved success probability when l < 3 + 2 2 ≈ 5.828 .
Quantum walks offer promising advantages for search algorithms over graphs. Among these, Grover’s quantum search provides a quadratic speedup with a time complexity of for unstructured search problems. Nevertheless, Grover search performs poorly on cyclic graphs due to the dynamics of the Grover walk. This study explores the behavior of Grover’s quantum walk on cyclic graphs , analyzing the probability distribution of finding the marked vertex. To analyze this behavior, we extend each vertex of the cycle by attaching semi-infinite-length paths (tails). We develop a direct analytical approach to obtain the transition matrix , whose elements are independent of . For small cycles , we observe success probabilities exceeding 0.5. In contrast, with large cycles, the effectiveness drops rapidly. We observe that the success probability approaches zero as the size of the graph increases , indicating the limitations of the quantum search in cyclic structures.
Graph partitioning is a crucial problem in computer science with applications in diverse areas like VLSI design, image segmentation, and network analysis. Traditional classical algorithms often face significant challenges when dealing with large and complex graphs, leading to increased computational time and memory requirements. This paper explores the potential of quantum walk algorithms to address this issue. We propose a framework for leveraging quantum walk algorithms to efficiently explore the connectivity structure of a graph and identify optimal partitions. The core idea involves mapping the graph's nodes and edges onto a quantum system and utilizing the inherent parallelism of quantum walks to traverse the graph. We present a theoretical analysis of the algorithm's potential speedup compared to classical approaches, focusing on scenarios where quantum computation offers a significant advantage. Specifically, we formulate the graph partitioning problem as a search problem on a graph represented as a quantum walk, considering the Hamiltonian operator and its eigenstates for efficient traversal. The effectiveness of the algorithm is dependent on the graph's structure and the ability to accurately simulate the quantum walk. The presented methodology provides a novel approach to graph partitioning, potentially leading to substantial performance improvements for large, complex graphs.
Abstract In this paper, we study Grover’s search algorithm focusing on continuous-time quantum walk on graphs. We propose an alternative optimization approach to Grover’s algorithm on graphs that can be summarized as follows: Instead of finding specific graph topologies convenient for the related quantum walk, we fix the graph topology and vary the underlying graph Laplacians. As a result, we search for the most appropriate analytical structure on graphs endowed with fixed topologies yielding better search outcomes. We discuss strategies to investigate the optimality of Grover’s algorithm and provide an example with an easy tunable graph Laplacian to investigate our ideas.
One of the most important algorithmic applications of quantum walks is to solve spatial search problems. A widely used quantum algorithm for this problem, introduced by Childs and Goldstone [Phys. Rev. A 70, 022314 (2004)], finds a marked node on a graph of nnn nodes via a continuous-time quantum walk. This algorithm is said to be optimal if it can find any of the nodes in O(n)O(\sqrt{n})O(n​) time. However, given a graph, no general conditions for the optimality of the algorithm are known and previous works demonstrating optimal quantum search for certain graphs required an instance-specific analysis. In fact, the demonstration of the necessary and sufficient conditions that a graph must fulfill for quantum search to be optimal has been a long-standing open problem. In this work we make significant progress towards solving this problem. We derive general expressions, depending on the spectral properties of the Hamiltonian driving the walk, that predict the performance of this quantum search algorithm provided certain spectral conditions are fulfilled. Our predictions are valid, for example, for (normalized) Hamiltonians whose spectral gap is considerably larger than ${n}^{\ensuremath{-}1/2}$. This allows us to derive necessary and sufficient conditions for optimal quantum search in this regime, as well as provide examples of graphs where quantum search is suboptimal. In addition, by extending this analysis, we are also able to show the optimality of quantum search for certain graphs with very small spectral gaps, such as graphs that can be efficiently partitioned into clusters. Our results imply that, to the best of our knowledge, all prior results analytically demonstrating the optimality of this algorithm for specific graphs can be recovered from our general results.
Kronecker graphs, obtained by repeatedly performing the Kronecker product of the adjacency matrix of an ``initiator'' graph with itself, have risen in popularity in network science due to their ability to generate complex networks with real-world properties. We explore spatial search by continuous-time quantum walk on Kronecker graphs. Specifically, we give analytical proofs for quantum search on first-, second-, and third-order Kronecker graphs with the complete graph as the initiator, showing that search takes Grover's O(N)O(\sqrt{N})O(N​) time. Numerical simulations indicate that higher-order Kronecker graphs with the complete initiator also support optimal quantum search.