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Hessian Calculations

We can define a search direction in the Newton Optimisation Method as:

x=-H^{-1}\nabla f

where H is the Hessian; a matrix of partial second order derivatives, and the \nabla f is the Jacobian matrix stacked into a column vector form.

The partial derivatives of the Hessian matrix can be calculated with the grape algorithm (Gradient Ascent Pulse Engineering)

\left\langle\sigma\right|\hat{\hat{U}}_N,\dots,\hat{\hat{U}}_{m+1}\frac{\partial^2}{\partial c_m^2}\hat{\hat{U}}_m,\dots,\hat{\hat{U}}_1\left|\psi_0\right\rangle