  In this work, we derive analytic formulae that determine the effect of error
mechanisms on 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 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 mixed state of the ion crystal's motion that is
diagonal in the Fock state basis. Our treatment of static motional frequency
shifts, trap anharmonicities, heating, and motional dephasing apply to both
laser-based and laser-free gates, while our treatment of field imhomogenieties
applies to laser-free systems.
