  One often needs to estimate how fast an evolving state of a quantum system
can depart from some target state or target subspace of a Hilbert space. Such
estimates are known as quantum speed limits. We derive a quantum speed limit
for a general time-dependent target subspace. When the target subspace is an
instantaneous invariant subspace of a time-dependent Hamiltonian, the obtained
quantum speed limit bounds the adiabatic fidelity, which is a figure of merit
of quantum adiabaticity. We also compare two states evolving under two
different Hamiltonians and derive a bound on the Loschmidt echo.
