Optimal control of the ground states of the quantum transverse field Ising model

We show that, by properly applying a time-dependent transverse field to the 1D Ising model (TFIM), one can always transform one ferromagnetic (FM) ground state to the other FM state with opposite spin orientation. A proof is established through an analysis of the dynamical Lie algebra generated by repeated commutators of the terms in the Hamiltonian. To find the control protocol that completes the transformation in the shortest possible time, we adopt the optimal control theory which provides an efficient numerical procedure to obtain a solution and examine its optimality. Depending on the maximally allowed controlled amplitude, the control profile can be either bang-bang or bang-singular bang. Moreover, both the optimal time and protocol exhibit a weak dependence on the system size. Inspired by the complete state transfer, the TFIM system may be used as a classical memory, where the two FM ground states represent a bit, and the bit flipping operation is accomplished by the optimal protocol.