From a few VQE tutorials online I see that they normally start with something like:
VQE is a way of getting a good estimate for the upper bound of the ground state of a quantum system's Hamiltonian. The Hamiltonian is known.
Then they proceed to show you a single qubit Hamiltonian as a 2x2 matrix, and go through the VQE algorithm to get the ground state energy (the smaller of the two eigenvalues).
Cool, but as devil's advocate, I could have just pulled out my high school maths to analytically get the smaller of the two eigenvalues. And with larger matrices, a classical computer will do the job well (which afaik is not a hard problem).
My question then is at what point does a Hamiltonian's matrix representation become so large, that even though we know it, we'd go to the trouble of probing the quantum system itself rather than solving for the eigenvalues on a classical computer? And how does that relate to the current and near future statuses of classical and quantum computing?
Please feel free to completely sidestep my question and nudge me back on track if I'm off-paradigm or started from the wrong premise.