I have been trying to implement a QAOA for solving a traveling salesman problem (TSP) using qulacs and python. However, even for 3 cities, the implementation fails. Within QAOA, we try to minimise $$ \begin{equation} F_p(\gamma,\beta) = \langle \gamma,\beta | C | \gamma,\beta\rangle, \end{equation} $$ where $C$ is the cost function of the TSP, and $|\gamma,\beta\rangle$ is a quantum state depending on these two angles. I had a closer look at my classical optimisation of the angles $\beta, \gamma$, for which I used the scipy.optimize.minimize function with the Nelder-Mead method. I realised that the resulting optimal angles are highly dependent on the initial angles. Additionally, I had a look at my cost function $C$. It seems like the optimisation got stuck in many local minima.

I have seen several implementations of a QAOA TSP using other software frameworks, and most of them also used scipy.optimize.minimize for the angles optimisation. Is getting stuck in local minima a known issue for QAOA TSP, or do I have to search for another error source? If the first, how can I overcome this issue?