Generalized geometric speed limits for quantum observables

Leveraging quantum information geometry, we derive generalized quantum speed limits on the rate of change of the expectation values of observables. These bounds subsume and, for Hilbert space dimension $\geq 3$, tighten existing bounds -- in some cases by an arbitrarily large multiplicative constant. The generalized bounds can be used to design "fast" Hamiltonians that enable the rapid driving of the expectation values of observables with potential applications e.g.~to quantum annealing, optimal control, variational quantum algorithms, and quantum sensing. Our theoretical results are supported by illustrative examples and an experimental demonstration using a superconducting qutrit. Possibly of independent interest, along the way to one of our bounds we derive a novel upper bound on the generalized quantum Fisher information with respect to time (including the standard symmetric logarithmic derivative quantum Fisher information) for unitary dynamics in terms of the variance of the associated Hamiltonian and the condition number of the density matrix.