A Cost Function Approach for Comparison of Harmonically Trapped Particle Transport Protocols

Transporting a particle in a harmonic potential between two locations arises naturally in array-based proposals for a scalable trapped ion quantum computer. Transport protocols parameterize the potential minimum location over time, and a protocol satisfying certain boundary conditions can ensure no excitation or local phase are incurred, which we call ideal transport. Ideal transport has an infinite solution space, but many solutions to ideal transport possess unrealizable changes in control field magnitudes or large excursions of the potential minimum beyond the interval of transport. We quantify deviations from a linear trajectory both in position and velocity, compute the overlap of the transported state with its translated initial state, and linearly combine these terms to construct a cost functional of the trajectory to compare the behavior of distinct quantum harmonic transport protocols without a full quantum simulation of particle displacement dynamics. We also show that not all terms can be simultaneously optimized in general, but rather that polynomial- and Fourier series-based trajectories exhibit different regions of superiority as quantified by the cost function.