Finding Small and Large $k$-Clique Instances on a Quantum Computer

Algorithms for triangle finding, the smallest nontrivial instance of the
$k$
-clique problem, have been proposed for quantum computers. Still, those algorithms assume the use of fixed access time quantum RAM. In this article, we present a practical gate-based approach to both the triangle-finding problem and its NP-hard
$k$
-clique generalization. We examine both constant factors for near-term implementation on a noisy intermediate scale quantum computing (NISQ) device and the scaling of the problem to evaluate long-term use of quantum computers. We compare the time complexity and circuit practicality of the theoretical approach and actual implementation. We propose and apply two different strategies to the
$k$
-clique problem, examining the circuit size of Qiskit implementations. We analyze our implementations by simulating triangle finding with various error models, observing the effect on damping the amplitude of the correct answer, and compare to execution on six real IBM quantum machines. Finally, we estimate the approximate quantum volume needed so that the smallest instance of our approach can be executable with minimal error on a real NISQ device.