  Efficient error estimates for the Trotter product formula are central in
quantum computing, mathematical physics, and numerical simulations. However,
the Trotter error's dependency on the input state and its application to
unbounded operators remains unclear. Here, we present a general theory for
error estimation, including higher-order product formulas, with explicit input
state dependency. Our approach overcomes two limitations of the existing
operator-norm estimates in the literature. First, previous bounds are too
pessimistic as they quantify the worst-case scenario. Second, previous bounds
become trivial for unbounded operators and cannot be applied to a wide class of
Trotter scenarios, including atomic and molecular Hamiltonians. Our method
enables analytical treatment of Trotter errors in chemistry simulations,
illustrated through a case study on the hydrogen atom. Our findings reveal: (i)
for states with fat-tailed energy distribution, such as low-angular-momentum
states of the hydrogen atom, the Trotter error scales worse than expected
(sublinearly) in the number of Trotter steps; (ii) certain states do not admit
an advantage in the scaling from higher-order Trotterization, and thus, the
higher-order Trotter hierarchy breaks down for these states, including the
hydrogen atom's ground state; (iii) the scaling of higher-order Trotter bounds
might depend on the order of the Hamiltonians in the Trotter product for states
with fat-tailed energy distribution. Physically, the enlarged Trotter error is
caused by the atom's ionization due to the Trotter dynamics. Mathematically, we
find that certain domain conditions are not satisfied by some states so higher
moments of the potential and kinetic energies diverge. Our analytical error
analysis agrees with numerical simulations, indicating that we can estimate the
state-dependent Trotter error scaling genuinely.
