  We provide state-dependent error bounds for strongly continuous unitary
representations of connected Lie groups. That is, we bound the difference of
two unitaries applied to a state in terms of the energy with respect to a
reference Hamiltonian associated to the representation and a left-invariant
metric distance on the group. Our method works for any connected Lie group and
the metric is independent of the chosen representation. The approach also
applies to projective representations and allows us to provide bounds on the
energy constrained diamond norm distance of any suitably continuous channel
representation of the group.
