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In this work, we derive analytic formulae that determine the effect of errormechanisms on one- and two-qubit gates in trapped ions and electrons. First, weanalyze, and derive expressions for, the effect of driving fieldinhomogeneities on one-qubit gate fidelities. Second, we derive expressions fortwo-qubit gate errors, including static motional frequency shifts, trapanharmonicities, field inhomogeneities, heating, and motional dephasing. Weshow that, for small errors, each of our expressions for infidelity convergesto its respective numerical simulation; this shows our formulae are sufficientfor determining error budgets for high-fidelity gates, obviating numericalsimulations in future projects. All of the derivations are general to anyinternal qubit state, and any mixed state of the ion crystal's motion that isdiagonal in the Fock state basis. Our treatment of static motional frequencyshifts, trap anharmonicities, heating, and motional dephasing apply to bothlaser-based and laser-free gates, while our treatment of field imhomogenietiesapplies to laser-free systems.

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