5.1. Variational method
We illustrate the variational method through both classical and hybrid-quantum methods. We compare VQE to the variational Monte Carlo method. Further, we also compare the results for VQE to the Quantum Phase Estimation (QPE) algorithm, which is not a variational method.
In this section, we cover the following topics:
- Section 5.1.1, The Rayleigh-Ritz variational theorem
- Section 5.1.2, Variational Monte Carlo methods
- Section 5.1.3, Quantum Phase Estimation (QPE)
- Section 5.1.4, Description of the VQE algorithm
5.1.1. The Rayleigh-Ritz variational theorem
The Rayleigh-Ritz variational theorem states that the expectation value of the Hamiltonian 
of a system with respect to the state of an arbitrary wave function (
) is always an upper bound to the exact ground state energy 
of the system it describes:
where 
generally represents time, spatial, and spin variables. This formula is not assuming any particular...
