Variational costate gradient for Hamiltonian quantum control ($\frac{\delta J}{\delta f(t)}=\frac{2}{\hslash }\mathrm{Im}⟨\chi (t)|H_1|\psi (t)⟩-\lambda f(t)$)

For $J=⟨\psi (T)|A|\psi (T)⟩-(\lambda /2){\int }_0^T{f}^{2}dt$ and $i\hslash \dot{\psi }=(H_0+f{H}_1)\psi$, enforce the state equation with $-2\mathrm{Re}\int ⟨\chi |\dot{\psi }+iH\psi /\hslash ⟩dt$. Variation gives the same Schrödinger equation for the costate, propagated backward from $\chi (T)=A\psi (T)$. Control variation gives the displayed real gradient. Forward state propagation, backward costate propagation, and a constrained gradient update form direct-adjoint looping. These are necessary stationarity equations, not a guarantee of a global optimum.