Numerous studies have investigated the simulation of an X-qubit circuit, demonstrating the 'classical simulation supremacy' on supercomputers. X is less than 50. However, useful VQE circuits are often long and require multiple iterations to converge. I would like to know the largest reported circuit, both in terms of qubit number and depth. My question does not concern VQE on a real device.
There was a paper came out from the quantum group at Google last year titled "Hartree-Fock on a superconducting qubit quantum computer". There they used 12 qubits which is probably one of the largest VQE demonstration on something that is physical realistic.
You can always design some ad hoc Hamiltonian that is large but really simple with a relatively simple ground state representation as well (so your VQE circuit or Ansatze doesn't need to be long) and perform your VQE. This would allow you to demonstrate a large VQE experiment but it would be unrealistic. Furthermore, if your Hamiltonian is simple and you also know ahead of time that the ground state is relatively simple, you don't need a quantum computer... you can do it efficiently classically. The point of QC is to look at problems where the ground state is highly superposition and entangled. This is where it would beat classical computer. This is why you keep seeing people talking about using quantum computer and performing VQE for those highly correlated systems...