= Gradient ascent pulse engineering
{title2=GRAPE}
= GRAPE
{c}
{synonym}
<Gradient ascent pulse engineering> parametrizes each control by time-slice amplitudes and optimizes a differentiable quantum-control objective. Forward products of slice <unitary operators> and backward adjoint propagation give derivatives with respect to every amplitude. For $U_j=e^{-iH_j\Delta t}$, the exact derivative is $\partial_{f_m}U_j=-i\int_0^{\Delta t}e^{-iH_j(\Delta t-\tau)}H_m e^{-iH_j\tau}\,d\tau$. Updating all slice amplitudes along these derivatives is gradient ascent for a fidelity, or gradient descent for its negative.
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