How to explain that I get a value lower than the smallest possible through minimization procedure in VQE?

Question

As far as I know after minimization I have to obtain a value $E_{0}\le \frac{\langle \psi (\theta)|H|\psi (\theta)\rangle}{\langle \psi (\theta)|\psi (\theta)\rangle}$, where $E_{0}$ - eigenvalue of ground state for hamiltonian $H$. Sometimes the algorithm give the value close to $E_{0}$, but far more often I get values lower than that.

I use hardwave efficient ansatz for initial state generation.

Hamiltonian consists of Pauli-strings $H=\sum_{ijkl}\sigma_i\sigma_j\sigma_k\sigma_l$.

For parameters optimization I use "COBYLA" and "Nelder-Mead" methods.

Could it be that the ansatz produce a state space which is not large enough?

Answer 1

The proof of the variational theorem (the theorem that the ground state energy is the lowest possible energy you can get from $\frac{\langle \psi|H|\psi\rangle}{ \langle \psi | \psi \rangle}$) is very simple: https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics)

If you get a lower energy, it means you don't actually have $\frac{\langle \psi|H|\psi\rangle}{ \langle \psi | \psi \rangle}$. For example if the quantum hardware doesn't give you $H|\psi\rangle$ but instead gives you $H|\psi\rangle + \epsilon |\psi\rangle $ where $\epsilon$ is some non-zero error, then when you plug everything into the proof of the variational theorem you may find that you are no longer guaranteed to always get energies equal to or higher than the ground state energy.