Polynomial scaling of QAOA for ground-state preparation of the fully-connected p-spin ferromagnet
Matteo M. Wauters, Glen Bigan Mbeng, Giuseppe E. Santoro · arXiv (Cornell University) · 2020
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 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 $2{\\rm P}$ is much smaller than the system size ${\\rm N}$, because of the large number of sub-optimal local minima. Instead, when ${\\rm P}$ exceeds a critical value ${\\rm P}^*_{\\rm N} \\propto {\\rm N}$, 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 ${\\rm N}$.