Evaluating the performance of the Quantum Approximate Optimisation Algorithm to solve the Quadratic Assignment Problem

The performance of the Quantum Approximate Optimisation Algorithm (QAOA) in solving the Quadratic Assignment Problem (QAP) is evaluated, with the Variational Quantum Eigensolver (VQE) as a benchmark. The QAP is directly revelant to numerous industry scenarios. The QAP, a Combinatorial Optimisation Problem (COP), is classified as
$\mathcal{NP}$
-Hard. This classification means CPU time increases exponentially as the problem size scales when solving the QAP using deterministic optimisation techniques. Therefore, this work investigates the QAOA in search of a non-deterministic optimisation technique to efficiently obtain solutions to the QAP. This research compares two warm start techniques to solve QAP instances of sizes 3 to 7. The metrics of comparison - that measure efficiency and solution quality - were introduced in previous work on this topic. For the QAOA, the impact of the p-value, a determination of circuit depth, is investigated. Of the two quantum hybrid heuristics, the VQE retrieves solutions in a shorter computational time with a smaller circuit size, which allows for solving instances with a larger problem size. Compared to the VQE, the QAOA performs better in terms of feasibility as the problem size scales. The quantum warm start method results implies that the QAOA may not maintain higher solution quality for instances larger than size 4. Still, further investigation should be conducted once quantum devices with more qubits and higher quantum volumes are available.