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Variational Quantum Eigensolver is a popular algorithm in Quantum Computing. But the ansatz part is very tricky. I do not really understand if they are built on some intuition, according to hardware or something else; or if it was just a trial and error approach.
What do you think about it?
VQE can be used for many things. The most popular application of VQE is for the quantum chemistry problem, as in this paper, where they are trying to find the ground state wavefunction of a molecular Hamiltonian (i.e. the VQE is trying to find the eigenvector with the smallest eigenvalue/energy). Here you can see that they suggest a unitary coupled cluster (UCC) ansatz. The reason they choose UCC is because it is well-known that coupled cluster already gives a very good approximation of the ground state wavefunction, in fact it is the basis for what chemists call the "gold standard of quantum chemistry".
Remember VQE is a heuristic. The better the ansatz that you start with, the more likely your VQE will perform well. As you correctly said in your question, you can use intuition, or trial-and-error, or just use any knowledge you have of the problem to come up with something that you believe will work well (as in the case of using a coupled cluster ansatz for the problem where coupled cluster is already considered "the gold standard" for people solving the problem on classical computers).
There is no general recipe for how to come up with the ansatz for VQE which will universally work well on every VQE problem, and that is why VQE is called a "heuristic".
