Circuit Design for Clique Problem and Its Implementation on Quantum
  Computer
Finding cliques in a graph has several applications for its pattern matching
ability. $k$-clique problem, a special case of clique problem, determines
whether an arbitrary graph contains a clique of size $k$, has already been
addressed in quantum domain. A variant of $k$-clique problem that lists all
cliques of size $k$, has also popular modern-day applications. Albeit, the
implementation of such variant of $k$-clique problem in quantum setting still
remains untouched. In this paper, apart from theoretical solution of such
$k$-clique problem, practical quantum gate-based implementation has been
addressed using Grover's algorithm. This approach is further extended to design
circuit for the maximum clique problem in classical-quantum hybrid
architecture. The algorithm automatically generates the circuit for any given
undirected and unweighted graph and any given $k$, which makes our approach
generalized in nature. The proposed approach of solving $k$-clique problem has
exhibited a reduction of qubit cost and circuit depth as compared to the
state-of-the-art approach, for a small $k$ with respect to a large graph. A
framework that can map the automated generated circuit for clique problem to
quantum devices is also proposed. An analysis of the experimental results is
demonstrated using IBM's Qiskit.
Arpita Sanyal Amit Saha Debasri Saha Banani Saha Amlan Chakrabarti