  Clique problem has a wide range of applications due to its pattern matching
ability. There are various formulation 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 in an
asymptotically faster manner by using quantum algorithms as compared to the
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 state-of-the-art quantum circuit for $k$-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).
