General Amplitude Modulation for Robust Trapped-Ion Entangling Gates

Trapped-ion systems are a promising route toward the realization of both near-term and universal quantum computers. One of the remaining challenges is improving the fidelity of two-qubit entangling gates. These operations are often implemented by addressing individual ions with laser pulses using the Molmer-Sorensen (MS) protocol. Amplitude modulation (AM) is a well-studied extension of this protocol, where the amplitude of the laser pulses is controlled as a function of time. The complexity of the pulse allows tradeoffs to be made between the laser power, gate time, and fidelity. We present an analytical study of AM using a Fourier series expansion so that the laser amplitudes may be represented as any continuous function in principle. We specifically study gate-timing errors, and we have shown that the sensitivity of the fidelity to these errors can be improved without a significant increase in the average laser power or the gate time. We plot atomic population vs time for both the MS protocol and the protocol with AM, highlighting the increased robustness of the AM gates. Furthermore, we numerically estimate the increased factor of power required to achieve a particular level of fidelity given an error estimate of the gate time fluctuations in the experimental setup.