Lower bounds and complexity dependence on locality and sparsity

Establish lower bounds for Hamiltonian learning from real-time evolution that clarify how the complexity of learning all parameters scales with the locality of the Hamiltonian and with the effective sparsity; in particular, ascertain whether dependence on effective sparsity is necessary and determine the optimal dependence.

Background

A lower bound of (1/ε) log(1/δ) is known for estimating a single parameter, but the overall complexity for learning all parameters remains unclear.

Understanding whether and how the complexity must intrinsically depend on locality and effective sparsity is important for characterizing fundamental limits of Hamiltonian learning algorithms.