Out-Of-Lamb-Dicke Error Mitigation for Trapped Ion Qudit Entanglement via Pulse Shaping

Trapped-ion qubits are a leading platform for quantum computation due to their long coherence times and high gate fidelities. Further, trapped ion qudits are an attractive method of scaling computation, both because they increase the size of the computational Hilbert space without requiring additional ions and because they interplay naturally with quantum error correction¹. However, we show that entangling gates with trapped ion qudits are more susceptible to errors due to higher-order motional terms neglected under the Lamb-Dicke approximation. These higher-order terms lead to residual spin-motion entanglement, reducing gate fidelity. In this work, we derive equations for the gate evolution to 3^rd order in the Lamb-Dicke approximation; we use these to create optimized gate profiles which reduce “Out-of-Lamb-Dicke” (OOLD) errors by orders of magnitude compared to rectangular pulses. We then apply these gate profiles to a pair of ⁴⁰Ca⁺ six-level qudits and demonstrate improvement relative to rectangular pulse profiles.

¹DeBry, et al. Error correction of a logical qubit encoded in a single atomic ion, arXiv:2503.13908 (2025).