I'm currently studying quantum computing and I have a question regarding the application of derivatives in the context of optimal control. I would like to provide a simple and generic example and would greatly appreciate any help. I've already tried discussing it with colleagues and professors, but I'm still unsure. I'm not aiming for mathematical rigor, I just want to be able to perform and understand the calculations.

Let's consider the expression $J = \langle\lambda(t)\rvert \mu(t)H_0 \lvert\psi(t)\rangle$, where $\lvert\psi(t)\rangle$ and $\lvert\lambda(t)\rangle$ are states in the Hilbert space, $H_0$ is a time-independent Hermitian operator, and $\mu(t)$ is an arbitrary function. I would like to know how to express $\frac{\partial J}{\partial \langle\lambda(t)\rvert}$, $\frac{\partial J}{\partial \lvert\psi(t)\rangle}$ and $\frac{\partial J}{\partial \lvert\lambda(t)\rangle}$ solely in terms of $\lvert\psi(t)\rangle$, $\lvert\lambda(t)\rangle$ (or duals), $H_0$, and $\mu(t)$. I understand that $\langle\lambda(t)\rvert$, $\lvert\psi(t)\rangle$ and $\lvert\lambda(t)\rangle$ are being treated as "independent variables" in the derivatives, similar to Lagrangian mechanics, but I'm having trouble grasping the entire process.



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