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

TL;DR

This paper derives analytic formulas to quantify the effects of various error mechanisms on the fidelity of one- and two-qubit gates in trapped ions and electrons, enabling precise error budgeting without extensive simulations.

Contribution

The authors present general analytic expressions for gate infidelities caused by motional errors, applicable to both laser-based and laser-free trapped ion systems, improving error analysis efficiency.

Findings

Analytic formulas match numerical simulations for small errors.

Formulas are applicable to any internal qubit state and motional mixed states.

Error budgets can be determined analytically, reducing reliance on simulations.

Abstract

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.