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Journal of Physics A Mathematical and Theoretical

Efficient quantum detection of maximum clique via hybrid heuristic and global information encoding

Wenmin Han, Shiqi Zheng, Peian Chen, Yukun Wang

Published May 13, 2026• 0 citations
Publisher

Abstract

Abstract The maximum clique problem (MCP) is to find the largest complete subgraph in an undirected graph, that is, the subgraph in which there are edges between every two different vertices. It is an NP-Hard problem with wide applications, including bioinformatics, social networks, data mining, and other fields. This paper proposes an enhanced quantum solution method that introduces a pre-detection phase based on a hybrid heuristic algorithm and a vertex pruning strategy, encoding the prior constraints on clique size into quantum registers as global information. It encodes the prior constraints on clique size into quantum registers as global information. In contrast, state-of-the-art Grover-based methods for n-vertex graphs require O ( n 2 n ) iterations and O ( n ) measurements. By leveraging a global information encoding mechanism, the proposed method compresses the number of iterations to O ( 2 n ) and only needs a constant number of measurements to identify all maximum cliques. We validate algorithmic correctness through simulations on IBM’s Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.

DOI
10.1088/1751-8121/ae6cc6

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Abstract The maximum clique problem (MCP) is to find the largest complete subgraph in an undirected graph, that is, the subgraph in which there are edges between every two different vertices. It is an NP-Hard problem with wide applications, including bioinformatics, social networks, data mining, and other fields. This paper proposes an enhanced quantum solution method that introduces a pre-detection phase based on a hybrid heuristic algorithm and a vertex pruning strategy, encoding the prior constraints on clique size into quantum registers as global information. It encodes the prior constraints on clique size into quantum registers as global information. In contrast, state-of-the-art Grover-based methods for n-vertex graphs require O ( n 2 n ) iterations and O ( n ) measurements. By leveraging a global information encoding mechanism, the proposed method compresses the number of iterations to O ( 2 n ) and only needs a constant number of measurements to identify all maximum cliques. We validate algorithmic correctness through simulations on IBM’s Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.
Abstract The maximum clique problem(MCP) is to find the largest complete subgraph in an undirected graph, that is, the subgraph in which there are edges between every two different vertices. It is an NP-Hard problem with wide applications, including bioinformatics, social networks, data mining, and other fields. This paper proposes an improved algorithm that dynamically tracks the maximum clique size by encoding prior constraints on the vertex count—derived from Turán's theorem and complete graph properties—into global variables through quantum circuit pre-detection. The algorithm further synergizes with Grover's search to optimize the solution space. Our auxiliary-qubit encoding scheme dynamically tracks clique sizes during quantum search, eliminating iterative measurements, achieving MCP solution with $O\left ( \sqrt{2^{n } } \right ) $ Grover iterations and $O\left ( 1 \right ) $ measurements. This represents an n\boldsymbol{n}n-fold improvement over state-of-the-art Grover-based methods, which require O(n2n)O(n\sqrt{2^n})O(n2n​) iterations and O(n)O(n)O(n) measurements for nnn-vertex graphs. We validate algorithmic correctness through simulations on IBM's Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.
The maximum clique of an undirected graph is the largest subgraph in which an edge exists between every vertex. The maximum clique problem presents itself in various fields and finding a tractable algorithm to solve the problem is important. This paper introduces an oracle to solve the maximum clique problem using Grover’s quantum algorithm. This approach potentially solves the maximum clique problem in O(∣V∣2∣V∣)O\left( {\left| V \right|\sqrt {{2^{\left| V \right|}}} } \right)O(∣V∣2∣V∣​) time complexity rather than the current classical time complexity of O(2|V|) of the best known algorithm, where |V| is the number of vertices in the graph. The full circuit implementation is presented using the open-source Qiskit environment [1]. The paper also analyzes the growth of the hardware resources requirements with the number of vertices in the targeted graph.
In the Maximum Clique Problem, the objective is to find a clique whose size is maximum among all the cliques of an arbitrary undirected and unweighted graph. Maximum Clique Problem resembles with minimum vertex cover problem, independent set problem. As it is an NP-hard problem, no polynomial time algorithms can be found. As these problems have several important practical applications such as information retrieval, community detection in network, spatial data mining etc, it is of great interest to try to synthesis the circuit of Maximum Clique Problem. In classical computing any brute-force solution to the Maximum Clique Problem requires an exponential increase of time with the size of the problem (i.e., with time complexity of O(2n)O(2^n)O(2n)). In this paper, Maximum Clique Problem has been solved using combinatorial approach of Grover's search algorithm. An algorithm has been proposed that auto-generates the circuit for any given undirected and unweighted graph which makes the approach generalized in nature for Maximum Clique Problem. Computational speed up is achieved by using our approach with the help of quantum mechanical effect that is quantum superposition. We have simulated the proposed circuit using IBM's QISkit platform and verified its correctness.
Maximum Clique Problem (MCP) is one of the most important NP-hard problems in the area of soft computing and it has many real world applications in numerous fields ranging from coding theory to the determination of the structure of a protein molecule. Different heuristic, Meta heuristic and hybrid solution approaches have been applied to obtain the solution. In this paper, we demonstrate a Quantum-inspired Evolutionary Algorithm (QEA) to solve MCP. We have used one dimensional arrays of Q-bits called Q-bit individuals to produce binary individuals. After production of binary individuals, we have repaired and improved them. Here, Q-gate is the main variation operator applied on Q-bit individuals. Our algorithm was tested on DIMACS benchmark graphs and 40 of them were tested. The results obtained here are extremely encouraging. For almost all of the datasets, we get the optimal results reported on DIMACS benchmark and also compared our results with other related works. For some cases we get better results than other works.
The maximal-clique problem, to find the maximally sized clique in a given graph, is classically an NP-complete computational problem, which has potential applications ranging from electrical engineering, computational chemistry, and bioinformatics to social networks. Here we develop a quantum algorithm to solve the maximal-clique problem for any graph GGG with nnn vertices with quadratic speedup over its classical counterparts, where the time and spatial complexities are reduced to, respectively, O(2n)O(\sqrt{{2}^{n}})O(2n​) and O(n2)O({n}^{2})O(n2). With respect to oracle-related quantum algorithms for the NP-complete problems, we identify our algorithm as optimal. To justify the feasibility of the proposed quantum algorithm, we successfully solve a typical clique problem for a graph GGG with two vertices and one edge by carrying out a nuclear magnetic resonance experiment involving four qubits.
Identifying a biclique with the maximum number of edges bears considerable implications for numerous fields of application, such as detecting anomalies in E-commerce transactions, discerning protein-protein interactions in biology, and refining the efficacy of social network recommendation algorithms. However, the inherent NP-hardness of this problem significantly complicates the matter. The prohibitive time complexity of existing algorithms is the primary bottleneck constraining the application scenarios. Aiming to address this challenge, we present an unprecedented exploration of a quantum computing approach. Efficient quantum algorithms, as a crucial future direction for handling NP-hard problems, are presently under intensive investigation, of which the potential has already been proven in practical arenas such as cybersecurity. However, in the field of quantum algorithms for graph databases, little work has been done due to the challenges presented by the quantum representation of complex graph topologies. In this study, we delve into the intricacies of encoding a bipartite graph on a quantum computer. Given a bipartite graph with n vertices, we propose a ground-breaking algorithm qMBS with time complexity O^*(2^(n/2)), illustrating a quadratic speed-up in terms of complexity compared to the state-of-the-art. Furthermore, we detail two variants tailored for the maximum vertex biclique problem and the maximum balanced biclique problem. To corroborate the practical performance and efficacy of our proposed algorithms, we have conducted proof-of-principle experiments utilizing IBM quantum simulators, of which the results provide a substantial validation of our approach to the extent possible to date.