A quantum walk-assisted approximate algorithm for bounded NP optimisation problems

This paper describes an application of the quantum approximate optimisation algorithm (QAOA) to efficiently find approximate solutions for computational problems contained in the polynomially bounded NP optimisation complexity class (NPO PB). We consider a generalisation of the QAOA state evolution to alternating quantum walks and solution-quality-dependent phase shifts and use the quantum walks to integrate the problem constraints of NPO problems. We apply the concept of a hybrid quantum-classical variational scheme to attempt finding the highest expectation value, which contains a high-quality solution. We synthesise an efficient quantum circuit for the constrained optimisation algorithm, and we numerically demonstrate the behaviour of the circuit with respect to an illustrative NP optimisation problem with constraints, minimum vertex cover. With examples, this paper demonstrates that the degree of accuracy to which the quantum walks are simulated can be treated as an additional optimisation parameter, leading to improved results.