Algebraic metrology: Nonoptimal but pretty good states and bounds

We investigate quantum metrology using a Lie algebraic approach for a class of Hamiltonians, including local and nearest-neighbor interaction Hamiltonians. Using this Lie algebraic formulation, we identify and construct highly symmetric states that admit Heisenberg scaling in precision for phase estimation in the absence of noise. For the nearest-neighbor Hamiltonian we also perform a numerical scaling analysis of the performance of pretty good states and derive upper bounds on the quantum Fisher information.