  We compute how phonon creation affects the fidelity of the quantum spin
dynamics in trapped ion simulators. A rigorous treatment of the quantum
dynamics is made by employing an exact operator factorization of the evolution
operator. Although it is often assumed that phonon creation modifies the
dynamics of the spin evolution, for an Ising spin-spin interaction in an
external magnetic field, phonons have \textit{no effect} on the probabilities
of spin product states measured in the direction of the Ising model axis.
Phonons play a much more important role in influencing the effective spin
dynamics for Heisenberg or XY model spin simulators or for other observables,
like witness operators in the Ising model.
