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International Journal of Quantum Information

Correlation between the continuous-time quantum walk and cliques in graphs and its application

Mingyou Wu, Xi Li, Zhihao Liu, Hanwu Chen

Published Apr 22, 2021• 1 citation
Publisher

Abstract

The continuous-time quantum walk (CTQW) provides a new approach to problems in graph theory. In this paper, the correlation between the CTQW and cliques in graphs is studied, and an approximate algorithm for the maximum clique problem (MCP) based on the CTQW is given. Via both numerical and theoretical analyses, it is found that the maximum clique is related to the transmission characteristics of the CTQW on some special graphs. For general graphs, the correlation is difficult to describe analytically. Therefore, the transmission characteristics of the CTQW are applied as a vertex selection criterion to a classical MCP algorithm and it is compared with the original algorithm. Numerous simulation on general graphs shows that the new algorithm is more efficient. Furthermore, an approximate MCP algorithm based on the CTQW is introduced, which only requires a very small number of searches with a high approximation ratio.

DOI
10.1142/s021974992150009x

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The continuous-time quantum walk (CTQW) provides a new approach to problems in graph theory. In this paper, the correlation between the CTQW and cliques in graphs is studied, and an approximate algorithm for the maximum clique problem (MCP) based on the CTQW is given. Via both numerical and theoretical analyses, it is found that the maximum clique is related to the transmission characteristics of the CTQW on some special graphs. For general graphs, the correlation is difficult to describe analytically. Therefore, the transmission characteristics of the CTQW are applied as a vertex selection criterion to a classical MCP algorithm and it is compared with the original algorithm. Numerous simulation on general graphs shows that the new algorithm is more efficient. Furthermore, an approximate MCP algorithm based on the CTQW is introduced, which only requires a very small number of searches with a high approximation ratio.
In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity O(N^6) in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.
In this work, we consider the application of continuous time quantum walking(CTQW) to the Maximum Clique(MC) Problem. Performing CTQW on graphs will generate distinct periodic probability amplitude for different vertices. We will show that the intensity of the probability amplitude at frequency indeed implies the clique structure of some special kinds of graph. And recursive algorithms with time complexity O(N5)O(N^5)O(N5) in classical computers for finding the maximum clique are proposed. We have experimented on random graphs where each edge exists with probabilities 0.3, 0.5 and 0.7. Although counter examples are not found for random graphs, whether these algorithms are universal is not known to us.
The maximum clique (MC) problem holds significance in network analysis. Quantum-based algorithms have recently emerged as promising approaches for this problem. However, these algorithms heavily depend on parameters of quantum system and vary significantly for different graphs. In order to tackle this issue, we initially demonstrate that continuous-time quantum walks (CTQW) driven by the Seidel matrix offer valuable insights into the clique structure of graphs, outperforming the CTQW driven by adjacency matrix. Specifically, we conduct an in-depth analysis for CTQW of 4 types of graphs, meticulously calculating the amplitudes associated with different vertices. Our findings consistently reveal that vertices belonging to MC exhibit the highest intensity at the largest frequency component of the probability amplitude for these types of graphs. Considering the varying intensities, we propose a parameter-independent algorithm for determining the MC. We compare our algorithm with a typical quantum-based algorithm, the results indicate that our algorithm exhibits greater stability.
We propose a novel heuristic quantum algorithm for the Minimum Vertex Cover (MVC) problem based on continuous-time quantum walks (CTQWs). In this framework, the coherent propagation of a quantum walker over a graph encodes its structural properties into state amplitudes, enabling the identification of highly influential vertices through their transition probabilities. To enhance stability and solution quality, we introduce a dynamic decoupling (``freezing'') mechanism that isolates vertices already selected for the cover, preventing their interference in subsequent iterations of the algorithm. The method employs a compact binary encoding, requiring only ⌈log⁡2(V)⌉\lceil \log_2 (V)\rceil⌈log2​(V)⌉ qubits to represent a graph with VVV vertices, resulting in an exponential reduction of quantum resources compared to conventional vertex-based encodings. We benchmark the proposed heuristic against exact solutions obtained via Mixed-Integer Linear Programming (MILP) and against established classical heuristics, including Simulated Annealing, FastVC, and the 2-Approximation algorithm, across Erdős--Rényi, Barabási--Albert and regular random graph ensembles. Our results demonstrate that the CTQW-based heuristic consistently achieves superior approximation ratios and exhibits remarkable robustness with respect to network topology, outperforming classical approaches in both heterogeneous and homogeneous structures. These findings indicate that continuous-time quantum walks, when combined with topology-independent decoupling strategies, provide a powerful paradigm for large-scale combinatorial optimization and complex network control, with potential applications spanning infrastructure resilience, epidemic containment, sensor network optimization, and biological systems analysis.
Determining the cardinality of the maximum clique on a simple graph is a NP-hard problem with several applications of interest. The best know classical algorithm for this problem has 𝒪(3𝑛/3) time complexity. Classical computing uses phenomena from classical physics to model, store and process information. Consequently, the entire model of classical computing is bounded superiorly by the limits of classical physics. As an alternative, Quantum Computing has been developed to take advantage of quantum phenomena and provide the constructions of circuits capable of performing certain computations more efficiently. In this paper, two quantum solutions to the maximum clique problem will be proposed and analyzed, based on the Grover Quantum Search algorithms and the Quantum Minimal Search Algorithm, comparing their respective time complexities with the best known classical solution.
In this dissertation we demonstrate that the continuous-time quantum walk models remain powerful for nontrivial graph structures. We consider two aspects of this problem. First, it is known that the standard Continuous-Time Quantum Walk (CTQW), proposed by Childs and Goldstone, can propagate quickly on the infinite path graph. However, the Schrödinger equation requires the Hamiltonian to be symmetric, and thus only undirected graphs can be implemented. In this thesis, we address the question, whether it is possible to construct a continuous-time quantum walk on general directed graphs, preserving its propagation properties. Secondly, the quantum spatial search defined through CTQW has been proven to work well on various undirected graphs. However, most of these graphs have very simple structures. The most advanced results concerned the Erdős-Rényi model of random graphs, which is the most popular but not realistic random graph model, and Barabási-Albert random graphs, for which full quadratic speed-up was not confirmed. In the scope of this aspect we analyze, whether quantum speed-up is observed for complicated graph structures as well.