Robust quantum circuit for clique problem with intermediate qudits

Clique problem has a wide range of applications due to its pattern matching ability. There are various formulations of clique problem like $k$-clique problem, maximum clique problem, etc. The $k$-Clique problem determines whether an arbitrary network has a clique or not whereas maximum clique problem finds the largest clique in a graph. It is already exhibited in the literature that the $k$-clique or maximum clique problem (NP-problem) can be solved asymptotically faster by using quantum algorithms compared to conventional computing. Quantum computing with higher dimensions is gaining popularity due to its large storage capacity and computation power. In this article, we have shown an improved quantum circuit implementation for the $k$-clique problem and maximum clique problem (MCP) with the help of higher-dimensional intermediate temporary qudits for the first time to the best of our knowledge. The cost of the state-of-the-art quantum circuit for the $k$-clique problem is colossal due to a huge number of $n$-qubit Toffoli gates. We have exhibited an improved cost and depth over the circuit by applying a generalized $n$-qubit Toffoli gate decomposition with intermediate ququarts (4-dimensional qudits).