One- and two-qubit gate infidelities due to motional errors in trapped ions and electrons*

In this work, we derive analytic formulae that determine the effect of error mechanisms in laser-free one- and two-qubit gates in trapped ions and electrons. First, we analyze, and derive expressions for, the effect of driving field inhomogeneities on one-qubit gate fidelities. Second, we derive expressions for two-qubit gate errors, including static motional frequency shifts, trap anharmonicities, field inhomogeneities, heating, and motional dephasing. We show that, for small errors, each of our expressions for infidelity converges to its respective numerical simulation; this shows that our formulae are sufficient for determining error budgets for high-fidelity gates, obviating numerical simulations in future projects. All of the derivations are general to any internal qubit state, and any \textit{mixed} state of the ion crystal's motion. Finally, we note that, while this manuscript focuses on laser-free systems, static motional frequency shifts, trap anharmonicities, heating, and motional dephasing are also important error mechanisms in laser-based gates, and our expressions apply.