  We study ultracold collisions of ions with neutral atoms in traps. Recently,
ultracold atom-ion systems are becoming available in experimental setups, where
their quantum states can be coherently controlled. This allows for an
implementation of quantum information processing combining the advantages of
charged and neutral particles. The state-dependent dynamics that is a necessary
ingredient for quantum computation schemes is provided in this case by the
short-range interaction forces depending on hyperfine states of both particles.
  In this work we develop a theoretical description of spin-state-dependent
trapped atom-ion collisions in the framework of a Multichannel Quantum Defect
Theory (MQDT) and formulate an effective single channel model that reduces the
complexity of the problem.
  Based on this description we simulate a two-qubit phase gate between a Ba135+
ion and a Rb87 atom using a realistic combination of the singlet and triplet
scattering lengths. We optimize and accelerate the gate process with the help
of optimal control techniques. Our result is a gate fidelity 0.999 within 350
microseconds.
