Polynomial scaling of QAOA for ground-state preparation: taming first-order phase transitions

We show that the quantum approximate optimization algorithm (QAOA) can construct with polynomially scaling resources the ground state of the fully-connected p-spin Ising ferromagnet, a problem that notoriously poses severe difficulties to a Quantum Annealing (QA) approach, due to the exponentially small gaps encountered at first-order phase transition for ${\rm p} \ge 3$. For a generic target state, we find that an appropriate QAOA parameter initialization is necessary to achieve a good performance of the algorithm when the number of variational parameters $2{\rm P}$ is much smaller that the system size ${\rm N}$, because of the large number of sub-optimal local minima. We find that when ${\rm P} > {\rm P}^*_{\rm N} \propto {\rm N}$, the structure of the parameter space simplifies, as all minima become degenerate. This allows to achieve the ground state with perfect fidelity with a number of parameters scaling extensively with ${\rm N}$, and with resources scaling polynomially with ${\rm N}$.