A method of detecting cliques in a graph includes determining, based on a number of nodes in the graph, a number of qubits to be included in a quantum processor. The method includes assigning to each node in the graph, a qubit of the quantum processor. The method includes operating on the qubits with a preparation circuit to create a quantum state in the qubits that corresponds to the graph. The method includes operating on the quantum state with a random walk circuit, and measuring the qubits of the quantum processor to detect cliques in the graph. The preparation circuit comprises a plurality of single- and two-qubit operators, wherein, for each pair of adjacent nodes in the graph, an operator of the plurality of two-qubit operators acts on a pair of qubits corresponding to the pair of adjacent nodes to create the quantum state.