Fig. 4: Relative and absolute error versus the number of iterations.

We implemented VQSE for states of: a n = 6, b n = 8, and c n = 10 qubits. In all cases, the ansatz for V(θ) was given by three layers of the Layered Hardware Efficient Ansatz of Fig. 3(b, top). Each curve represents the absolute or relative error (denoted Abs error or Rel error, respectively) of (18) obtained by training V(θ) when employing an adaptive, fixed-local, or fixed-global Hamiltonian. The number of iterations was 330 for (a) and 360 for (b) and (c). For the adaptive runs, we employed Algorithm 1, with the Hamiltonian being updated every 30 iterations. In each case, the adaptive approach performs the best as it achieves the smallest errors.