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What would be a possible ansatz quantum state in VQE (variational quantum eigensolver [1]) that would demonstrate the quantum advantage of VQE over classic computers? More specifically, I see that VQE has a quantum advantage if my ansatz state is created using some oracle function like in Grover's algorithm (because such a quantum state cannot be presented efficiently a classic computer). However, I am not sure if such a state would classify as an ansatz because it may come with stricter assumptions than VQE wrt quantum noise, coherence times etc. (which are usually the arguments in favor of VQE over other quantum algorithms like Grover or quantum phase estimation).

More detailed: Assume we want to efficiently simulate VQE on a classic computer. Using the same assumptions and notation as in [1], it suffices to show that vectors of the form $\langle \psi \mid \sigma^i_\alpha \sigma^j_\beta \mid \psi \rangle$ can be evaluated efficiently, where $\sigma^i_\alpha$ denote Pauli operators, while $\mid \psi \rangle$ is an ansatz quantum state. If $\mid \psi \rangle$ is nice enough (e.g., $H \mid 0 \rangle$), the expression involving the Pauli expression from above can be computed efficiently too. Consequently, if my ansatz is sufficiently nice, it seems that VQE can be simulated efficiently on a classic computer.

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