Variational Method

For a system described by an Hamiltonian $H$, we find an approximation to the Ground state Energy $E_0$ with the steps:

1. Parameterise quantum states as $|{\psi }_{\theta }⟩$.
2. Describe the expected Energy for the state $|{\psi }_{\theta }⟩$ as $E_{\theta }:=\frac{⟨{\psi }_{\theta }|H|{\psi }_{\theta }⟩}{⟨{\psi }_{\theta }|{\psi }_{\theta }⟩}$.
3. Find the minimal Energy by solving $0=\frac{\partial }{\partial \theta }{E}_{\theta }$.

This concept is foundational for the Variational Quantum Eigensolver algorithm.