  The intractability of deterministic solutions in solving $\mathcal{NP}$-Hard
Combinatorial Optimisation Problems (COP) is well reported in the literature.
One mechanism for overcoming this difficulty has been the use of efficient COP
non-deterministic approaches. However, with the advent of quantum technology,
these modern devices' potential to overcome this tractability limitation
requires exploration. This paper juxtaposes classical and quantum optimisation
algorithms' performance to solve two common COP, the Travelling Salesman
Problem (TSP) and the Quadratic Assignment Problem (QAP). Two accepted
classical optimisation methods, Branch and Bound (BNB) and Simulated Annealing
(SA), are compared to two quantum optimisation methods, Variational Quantum
Eigensolver (VQE) algorithm and Quantum Approximate Optimisation Algorithm
(QAOA). These algorithms are respectively executed on both classical devices
and IBM's suite of Noisy Intermediate-Scale Quantum (NISQ) devices. We have
encoded the COP problems for the respective technologies and algorithms and
provided the computational encodings for the NISQ devices. Our experimental
results show that current classical devices significantly outperform the
presently available NISQ devices, which agrees and extends with findings in the
literature. Further, we introduce additional performance metrics to better
compare the two approaches concerning computational time, feasibility and
solution quality. Our results show that the VQE performs better than QAOA for
these metrics, and we infer that this is due to the increased number of
operations required. Additionally, we investigate the impact of a new set of
basis gates on the quantum optimisation techniques and show they yield no
notable improvement in the results. Finally, we present the shortcomings of
state-of-the-art NISQ IBM quantum devices and argue for potential future work
and investigation.
