Testing Hardness in the Transverse-Field Ising Model using the Perturbed Ferromagnetic Chain*

Probing the role of coherence in quantum annealing relies on the use of “toy” Hamiltonians, whereby the distribution of computational states at the end of an anneal depends on the degree of coherence in the experimental implementation. Increasing the hardness generally relies on increasing the system size, and therefore the solution cannot be computed after the problem exceeds a few tens of qubits.

In this work we propose a new Hamiltonian, the perturbed ferromagnetic chain, where the degree of hardness can be controlled by a tuneable parameter at fixed system size.  We study the properties of the perturbed ferromagnetic chain and show that it possesses a false minimum and an exponentially large (in system size) first-excited-state manifold. We simulate both classical and open quantum system dynamical models and demonstrate that the properties of the perturbed ferromagnetic chain result in orders of magnitude difference in the ground state probability between the models.