  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 (QRAM). 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 computer (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 IBMQ machines. Finally, we
estimate the date when the methods proposed can run effectively on an actual
device based on IBM's quantum volume exponential growth forecast and the
results of our error analysis.
