Entangling an arbitrary pair of qubits in a long ion crystal

It is well established that the collective motion of ion crystals can be used as a quantum bus for multi-qubit entanglement. However, as the number of ions increases, it becomes difficult to directly entangle ions far apart and resolve all motional modes of the ion crystal. We introduce a scalable and flexible scheme for efficient entanglement between any pair of ions within a large ion chain, using an evenly distributed 50-ion crystal as an example. By performing amplitude and frequency modulation, we find high-fidelity pulse sequences that primarily drive a transverse motional mode with a wavelength of 4 ion spacings. We present two $500 \mu s$ pulses that can in theory suppress gate errors due to residual motion to below $10^{-4}$, and observe a trade-off between gate power and robustness against unwanted frequency offsets.