Optimizing Pulse Shapes of an Echoed Conditional Displacement Gate in a Superconducting Bosonic System

Echoed conditional displacement (ECD) gates for bosonic systems have become the key element for real-time quantum error correction beyond the break-even point. These gates are characterized by a single complex parameter β, and can be constructed using Gaussian pulses and free evolutions with the help of an ancillary transmon qubit. We show that there is a lower bound for the gate time in the standard construction of an ECD gate. We present a method for optimizing the pulse shape of an ECD gate using a pulse-shaping technique subject to a set of experimental constraints. Our optimized pulse shapes remain symmetric, and can be applied to a range of target values of β by tuning only the amplitude. We demonstrate that the total gate time of an ECD gate for a small value of β can be reduced either by relaxing the no-overlap constraint on the primitives used in the standard construction or via our optimal-control method. We show a slight advantage of the optimal-control method by demonstrating a reduction in the preparation time of a |+Z_GKP> logical state by thicksim10\%.