Optimal control of entangling operations for trapped ion quantum   computing
  Optimal control techniques are applied for the decomposition of unitary quantum operations into a sequence of single-qubit gates and entangling operations. To this end, we modify a gradient-ascent algorithm developed for systems of coupled nuclear spins in molecules to make it suitable for trapped ion quantum computing. We decompose unitary operations into entangling gates that are based on a nonlinear collective spin operator and complemented by global spinflip and local light shift gates. Among others, we provide explicit decompositions of controlled-NOT and Toffoli gates, and a simple quantum error correction protocol. 
V. Nebendahl, H. Haffner, and C. F. Roos