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In VQE we wish to minimize some cost function $F(\vec{x})$ that is dependent on a quantum state $\left| \psi_\vec{x} \right>$ which is prepared by a unitary $U(\vec{x})$ depending on some (typically real) parameters $\vec{x} \in \mathbb{R}^n$. A typical example for the cost function is the energy expectation value of some hamiltonian, i.e. $F(\vec{x}) = \left< \psi_\vec{x} \right|H \left|\psi_\vec{x} \right>$. Usually I can't measure my cost function directly, but only learn about it by making more and more measurements. For optimization I need some accuracy to decide in which direction to take the next optimization step, but I don't want to take arbitrarily many samples before moving on, since each sample means preparing $\psi_\vec{x}$ and measuring its energy w.r.t $H$.

Is there any research done, on what are optimal (in the sense of minimizing the number of calls to my quantum computer) sampling strategies for this kind of problem? Or what should I search for? "optimizing functions from samples" doesn't get me anywhere.

Cheers, Jan Lukas

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