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There exists a famous result from Google that the gradients of the parameters of quantum neural networks (QNN) vanish exponentially with the number of qubits in the quantum circuit. Their result hinges on sampling random parameterized unitary matrices from the Haar measure. In this setting, QNN's are untrainable at large numbers of qubits.

My question is, does there exist a similar result for entanglement? I.e, given the $\alpha$-Renyi entropy of a subsystem $k$ from the partial trace of a pure state density matrix $\rho_k = \text{Tr}_l[\rho]$, $l + k = n$, setting $\rho \equiv \rho(\theta) = U(\theta)|0\rangle^{\otimes n}\langle0|^{\otimes n} U^{\dagger}(\theta)$,

$$ S_{\alpha}(\rho_k(\theta)) = \frac{1}{1 - \alpha} \text{log}(\text{Tr}[\rho_k^{\alpha}(\theta)]) $$

is there a known result for

$$ \text{Var}[\partial_\theta S_\alpha(\rho_k(\theta))] $$

where, like Google paper, unitary matrices are sampled from the Haar measure? Is there any other result which suggests that entanglement is trainable? The main goal here is to determine if it is feasible given a general QNN to efficiently maximize the entanglement in an optimization procedure.

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