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

Abstract

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.

Appendix A: Numerical results for random graphs

Classical simulations of the quantum state evolution were performed to evaluate the quality of approximate solutions in the context of minimum vertex cover. We define the approximation quality for a particular problem instance as the ratio of the number of vertices in the minimum vertex cover to the approximate cover. Since for a n-qubit quantum register the classical computer must store all $2^n$ quantum amplitudes in memory, results were obtained for only low-n simulations ($\lesssim 20$). To reduce the computational cost, the state evolution is carried out by direct computation and multiplication of the matrices $e^{-i \gamma {\hat{C}}}$ and $e^{-i \beta {\hat{B}}}$, rather than simulating the quantum circuit. As in Sect. 5, the Nelder–Mead nonlinear optimiser [36] is used to optimise the expectation value $F_p(\vec {\beta }, \vec {\gamma })$.

The performance of the $p=2$ algorithm was tested on a random sample of G(n, 0.5) Erdős–Rényi graphs. The G(n, 0.5) Erdős–Rényi random graph model [37] has equal probability to select each of the $2^{n(n-1)/2}$ n-vertex graphs, so it gives a good impression of the ‘average case’ performance of the heuristic. The solution qualities for each random graph are shown in Fig. 10, with 20 instances considered per n. Taking $n=5$ as an example, the optimal vertex cover is found for all but two random instances tested. The solution quality decreases reasonably slowly, and for all trialled graphs the produced solution used at most 1.6 times the number of vertices as the optimal solution.

Fig. 10

Ratio of the number of vertices in the approximate cover to the minimum vertex cover. The grey line is the mean, and the shaded region is the 95% CI. The size of each data point is proportional to the number of tested instances with the same approximation quality

Note that the results in this ‘Appendix’ are an exact simulation of the desired QAOA state evolution, rather than a simulation of the quantum circuit. Hence, the results here are specific to the case $m \rightarrow \infty$, where m is the ‘walk simulation accuracy’ parameter introduced in Sect. 4.3. The solution qualities would be further improved by optimising over this parameter.