QAOA Belongs into VQE or the other way around?

Question

I have been reading a couple of papers in the arxiv and wanted to get a clarification regarding the relation between these two methods; is one a subset of the other?

Answer 1

QAOA belongs to VQE. Indeed, the idea of VQE is to use a parametrized quantum circuit $U(\theta)$ to minimize $$\langle 0|U(\theta)^{\dagger}H_PU(\theta)|0\rangle$$ in order to obtain an approximation of the groundstate of $H_P$. The circuit $U(\theta)$ is called an ansatz, and QAOA uses a particular type of ansatz that can be written $$U(\theta)=e^{i\beta_1H_P} e^{i\gamma_1H_M}...e^{i\beta_pH_P} e^{i\gamma_pH_M}$$ where $\theta=(\gamma_1,\beta_1,...,\gamma_p,\beta_p)$. Therefore, it can be seen as a special case of VQE