Structured light for addressable two-qubit gates in medium-sized ion crystals: modeling motional and dephasing errors
Geometric phase gates with addressed laser radiation are widely used in trapped-ion quantum computers to drive high-fidelity entangling gates on an arbitrary qubit pair of a larger ion chain. Such programmable connectivity exploits the collective phonons to mediate the desired interaction, while simultaneously achieving robustness to their thermal fluctuations by a careful calibration of the gate time with respect to the laser and vibrational-mode frequencies. Crosstalk errors can arise when the spectator qubits, which should ideally remain idle during the gate, are partially driven by the laser fields. The minimal crosstalk, which is set by the laser beam size on the spectator ions, can be achieved when the addressing beams propagate perpendicular to the ion chain. However, this results in exciting the radial phonons, and can complicate achieving high-fidelities as the ion chain grows due to the so-called spectral mode crowding. In this work, we present the theoretical description of a scheme recently implemented by Faorlin et $al.$. This scheme overcomes these complications by engineering addressable state-dependent forces along the ion-chain with laser beams that propagate perpendicular to it through the use of structured light gradients. In this way, one can excite the axial phonons and avoid the spectral mode crowding, while simultaneously minimizing the crosstalk. We present a detailed derivation of the entangling gate scheme, together with analytical closed-form expressions for the leading contribution to the gate error caused by thermal motional fluctuations and dephasing noise. Our results confirm that such fluctuations account for up to 93$\%$ of the error budget under the operating conditions of the gate implemented by Faorlin et $al.$. We identify target parameters to reduce the motional error from the reported $10^{-3}$ to the $10^{-5}$ level.