Polynomial scaling of QAOA for ground-state preparation of the fully-connected p-spin ferromagnet

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 p 3. For a target ground state at arbitrary transverse field, 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 P is much smaller than the system size N, because of the large number of sub-optimal local minima. Instead, when P exceeds a critical value P^* ₍ 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 N, and with resources scaling polynomially with N.