  Quadratic convergence throughout the active space is achieved for the
gradient ascent pulse engineering (GRAPE) family of quantum optimal control
algorithms. We demonstrate in this communication that the Hessian of the GRAPE
fidelity functional is unusually cheap, having the same asymptotic complexity
scaling as the functional itself. This leads to the possibility of using very
efficient numerical optimization techniques. In particular, the Newton-Raphson
method with a rational function optimization (RFO) regularized Hessian is shown
in this work to require fewer system trajectory evaluations than any other
algorithm in the GRAPE family. This communication describes algebraic and
numerical implementation aspects (matrix exponential recycling, Hessian
regularization, etc.) for the RFO Newton-Raphson version of GRAPE and reports
benchmarks for common spin state control problems in magnetic resonance
spectroscopy.
