Measuring the expectation value of a hamiltonian is an essential step in some algorithms like QAOA.
I noticed that the procedure always starts with decomposing the hamiltonian into a sum of Pauli strings, then proceeding to building separate circuits to measure the expectation value, then adding them all.
It seems like it would be simpler to simply measure the state, compute the hamiltonian on a classical computer for the bit string measured, repeat and take the average.
Is there a reason why we like to measure the expectation value using the Pauli decomposition method instead? I'm guessing the answer would be it's more efficient. But I can't see why or how. It seems like we need to repeat many measurements for both procedures.