Abstract
Finding cliques in a graph has a wide range of applications due to its pattern matching ability. The k-clique problem, a subset of the clique problem, determines whether or not an arbitrary network has a clique of size k. Modern-day applications include a variation of the k-clique problem that lists all cliques of size k. However, the quantum implementation of such a variation of the k-clique problem has not been addressed yet. In this work, apart from the theoretical solution of such a k-clique problem, practical quantum-gate-based implementation has been addressed using Grover's algorithm. In a classical-quantum hybrid architecture, this approach is extended to build the circuit for the maximum clique problem. Our technique is generalised since the program automatically builds the circuit for any given undirected and unweighted graph and any chosen k. For a small k with regard to a big graph, the proposed solution to addressing the k-clique issue has shown a reduction in qubit cost and circuit depth when compared to the state-of-the-art approach. A framework is also presented for mapping the automated generated circuit for clique problems to quantum devices. Using IBM's Qiskit, an analysis of the experimental results is demonstrated.
Highlights
Quantum computers were proposed in the early 1980s and the description of quantum mechanical computers was formalized in the late 1980s
An automated and generalized approach of quantum circuit synthesis for k-clique problem using Grover’s algorithm n with O( k ) iterations for any given undirected and unweighted graph has been proposed in this paper, which m gives all cliques of size k as output
We exhibited that the proposed approach of solving k-clique problem is cost efficient with respect to qubit and circuit depth, when k is very small with respect to a large graph
Summary
Quantum computers were proposed in the early 1980s and the description of quantum mechanical computers was formalized in the late 1980s. For example Shor’s Algorithm [1] for factoring integers, Grover’s Algorithm [2] for searching an unstructured database, Triangle finding by Magniez et al [3], Matrix Product Verification [4] have already been proposed and shown asymptotic improvements than their classical counterparts. In this paper, another computationally NP-problem i.e., clique problem [5] has been addressed in quantum setting.
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