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Efficient error estimates for the Trotter product formula are central inquantum computing, mathematical physics, and numerical simulations. However,the Trotter error's dependency on the input state and its application tounbounded operators remains unclear. Here, we present a general theory forerror estimation, including higher-order product formulas, with explicit inputstate dependency. Our approach overcomes two limitations of the existingoperator-norm estimates in the literature. First, previous bounds are toopessimistic as they quantify the worst-case scenario. Second, previous boundsbecome trivial for unbounded operators and cannot be applied to a wide class ofTrotter scenarios, including atomic and molecular Hamiltonians. Our methodenables 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-momentumstates of the hydrogen atom, the Trotter error scales worse than expected(sublinearly) in the number of Trotter steps; (ii) certain states do not admitan advantage in the scaling from higher-order Trotterization, and thus, thehigher-order Trotter hierarchy breaks down for these states, including thehydrogen atom's ground state; (iii) the scaling of higher-order Trotter boundsmight depend on the order of the Hamiltonians in the Trotter product for stateswith fat-tailed energy distribution. Physically, the enlarged Trotter error iscaused by the atom's ionization due to the Trotter dynamics. Mathematically, wefind that certain domain conditions are not satisfied by some states so highermoments of the potential and kinetic energies diverge. Our analytical erroranalysis agrees with numerical simulations, indicating that we can estimate thestate-dependent Trotter error scaling genuinely.

Strong Error Bounds for Trotter & Strang-Splittings and Their Implications for Quantum Chemistry