  We have recently demonstrated supervised deep learning methods for rapid
generation of radiofrequency pulses in magnetic resonance imaging
(https://doi.org/10.1002/mrm.27740, https://doi.org/10.1002/mrm.28667). Unlike
the previous iterative optimization approaches, deep learning methods generate
a pulse using a fixed number of floating-point operations - this is important
in MRI, where patient-specific pulses preferably must be produced in real time.
However, deep learning requires vast training libraries, which must be
generated using the traditional methods, e.g. iterative quantum optimal control
methods. Those methods are usually variations of gradient descent, and the
calculation of the fidelity gradient of the performance metric with respect to
the pulse waveform can be the most numerically intensive step. In this
communication, we explore various ways in which the calculation of fidelity
gradients in quantum optimal control theory may be accelerated. Four
optimization avenues are explored: truncated commutator series expansions at
zeroth and first order, a novel midpoint truncation scheme at first order, and
the exact complex-step method. For the spin systems relevant to MRI, the
first-order truncation is found to be sufficiently accurate, but also up to
five times faster than the machine precision gradient. This makes the
generation of training databases for the machine learning methods considerably
more realistic.
