Researchers Quantum Leap In Solving Maximum Clique Problem
Maximum Clique Problem A New Algorithm Solves Maximum Clique Problem with Unprecedented Quantum Leap Efficiency
Yukun Wang, Wenmin Han, Shiqi Zheng, and Peian Chen made a quantum computing breakthrough. Their novel Maximum Clique Problem (MCP) approach is much more efficient. An enhanced system for dynamically tracking prospective clique sizes solves this computationally expensive problem, which is vital in many scientific and commercial fields, with n times the efficiency of Grover-based alternatives. The Maximum Clique Problem seeks the largest "complete subgraph" in an undirected graph, a set of vertices with edges connecting each pair. The computational complexity of this problem increases exponentially with graph size, making classical solutions impossible for larger networks. This problem is NP-Hard. Classical exact algorithms, which often use Branch-and-Bound (B&B) techniques, have a worst-case time complexity of, where ‘n’ is the number of vertices. These approaches are unsuitable for large-scale networks because their complexity grows rapidly, making it hard to get good approximation ratios in polynomial time. Data mining, social network analysis, bioinformatics, and communication signal processing are among its many applications. It helps determine the largest group of persons all members know in social networks, providing significant community structure information. Quantum computing has long been considered a feasible solution to traditional approaches' shortcomings for some tough problems. Grover's approach offers a quadratic speedup for NP-complete tasks and is essential to quantum search. Earlier attempts to apply Grover's approach to MCP faced serious problems. Techniques typically required O(n) measurements and up to O(n√2^n) iterations. Weakness of the quantum circuit to dynamically access global clique size information during execution caused this inefficiency. Quantum states cannot reveal intermediate clique sizes without measurement, unlike classical algorithms that can adjust their search based on a dynamic ‘k’ (vertex count) parameter. This forced earlier + to perform numerous iterative full Grover searches, updating ‘k’ only after measurements, resulting in many total measurements. Dynamic quantum solution emerges The innovation is the new algorithm's Pre-Detection and Encoding method, which effectively avoids these limits. Auxiliary qubits encode prior vertex count limits onto global variables to dynamically track the maximum clique size. Using this inventive strategy, the MCP solution may be achieved with only O(√2^n) Grover iterations and O(1) measurements, eliminating the need for iterative measurements. This is n-fold better than Grover-based methods. The approach is organised around four steps: Quantum Pre-Detection and Encoding (QPDE), Cliques Detector, MCP Detector, and Diffusion.
Quantum pre-detection and encoding This crucial first step obtains and pre-stores the vertex number information of the largest clique, max_c, in a quantum register. The QPDE stage uses MCP Prior Constraints Acquisition to simplify compute. This applies Turán's theorem and complete graph characteristics. After exactly initialising the maximum clique size into the quantum register, theoretical limits drastically decrease the range of feasible values. Depending on the number of vertices and edges, Turán's theorem limits clique size lower than complete graph properties. This precise constraint acquisition reduces the search space for quantum computing. Quantum MCP Size Detection finds cliques in the improved vertex number range using quantum circuits. List all potential vertex sets, count r-cliques with a simplified quantum counter, and determine clique formation with multi-controlled Toffoli (MCT) gates. A matching qubit in max_c register is set to |1⟩ when an r-clique is found. A network with five vertices would indicate a maximum clique size of 4. Cliques Detector After initialising vertex qubits into uniform superposition with Hadamard gates, the Cliques Detector discovers every clique in the graph. Each phrase determines if a pair of vertices and their related edge match clique restrictions using a conjunctive normal form (CNF). These clauses' truth values are stored in auxiliary registers and implemented using 3-controlled and X gates. A multi-controlled X gate performs the conjunctive operation on all clauses, marking cliques by flipping a target bit to |1⟩ only when all clause criteria are met. MCP Detector Amplitude amplification requires the MCP Detector to find the greatest clique and phase invert its quantum states. It builds on Cliques Detector and QPDE output. Consistency comparison, sorting, and quantum state duplication occur here. The QPDE stage's max_c information is compared to the putative clique's sorted bit sequence using XNOR operations. If both sequences are identical, a Z-gate operation inverts the output qubit phase to designate the largest clique. Diffusion Operator Grover iterations end with the diffusion operator. This component selectively increases the amplitudes of marked target states (negative phases) while suppressing non-target states. We find the solution when the system collapses into one of the maximum clique states with high probability after an ideal number of iterations and the target states' measurement probability is almost unity. Quantifying Benefits and Future Prospects IBM's Qiskit platform simulations verified the algorithm's accuracy. Its 96% success rate for a 4-vertex graph is comparable to previous Grover-based methods. Significant quantum resource utilisation benefits from the provided algorithm. QPDE has a gate complexity of, although it is only performed once during the search process, reducing its impact. Gate complexity remains competitive. The worst-case quantum bit complexity outperforms Matheus and Haverly. Due to its broader solution space, the Arpita technique has more Grover iterations and a somewhat lower success probability while having less qubit. This unique algorithm offers a potential approach to handle increasingly complicated network research, especially for large-scale graphs where standard solutions are computationally prohibitive. Research will focus on improved encoding, circuit design optimisation, and alternative QPDE approaches to reduce quantum resource overhead and processing cost.
