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. However, one of the pressing 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. We present an analytical study of the effect of gate-timing errors on the fidelity of MS gates with AM, using a Fourier series expansion to maintain the generality of the laser amplitude's functional form. Imposing conditions on these Fourier coefficients produces trade-offs between the laser power, gate time, and fidelity. Numerical optimization is then employed to identify the minimum-power pulse at a given level of fidelity. Our central result is that we improve the leading order dependence on gate timing errors from O(Delta t^2) to O(Delta t^6) with the addition of one linear constraint on the Fourier coefficients. This occurs without a significant increase in the average laser power or the gate time. A set of constraints is also presented that can be used, in principle, to achieve arbitrarily high orders of insensitivity to gate-timing errors.