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Suppose I have a cost function $C = \langle \psi \rvert U^\dagger O U \rvert \psi \rangle$ for a fixed observable $O$ and a fixed state $\rvert \psi \rangle$. I know that usually people take the derivative of the cost function wrt parameters, but here I want to take the derivative of $C$ wrt $U$: $\frac{\partial C}{\partial U} = \langle \psi \rvert \frac{\partial U^\dagger}{\partial U} O U \rvert \psi \rangle + \langle \psi \rvert U^\dagger O \rvert \psi \rangle$, and I am not sure how to deal with $\frac{\partial U^\dagger}{\partial U}$ part. Any help would be appreciated!

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    $\begingroup$ What is the origin of the unitary matrix and how can it change? (I hope you're not allowing an entirely arbitrary $n$-qubit unitary!) $\endgroup$
    – DaftWullie
    Commented Jul 3, 2023 at 6:13

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If I take your questions literally, then the answer is fairly straightforward: $$ \frac{dU^\dagger}{dU}=-1. $$ You should think about it this way: we know $UU^\dagger=I$. If $U$ changes by $\delta U$ (small), $U^\dagger$ must change to compensate: $$ (U+\delta U)(U-\delta U)=I+O(\delta U^2) $$ Thus, to first order, $U^\dagger$ changes to $U^\dagger-\delta U$.

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