diff --git a/README.md b/README.md index a0540f7..375ae96 100644 --- a/README.md +++ b/README.md @@ -43,6 +43,38 @@ can be applied to best run quantum optimization algorithms on real hardware: 3. Application of SWAP strategies -> `swap_strategies` 4. QAOA cost function -> `cost_function` +## QAOA Formulation Convention + +The convention in this repository is that the (depth-one) QAOA should create an Ansatz of the form: + +$$ +\begin{align} +\exp\left(-i\beta H_m\right)\exp\left(-i\gamma H_c\right)\vert+\rangle^{\otimes n} +\end{align} +$$ + +to **minimize** the energy of $H_c$. Since we are minimizing $\\langle H_c\\rangle$, +the initial state of the ansatz should be the ground state of the mixer $H_m$. This enforces: + +$$ +\\begin{align} +H_m=-\\sum_iX_i. +\\end{align} +$$ + +As cost operator we apply: + +$$ +\\begin{align} +H_c=\\sum_{i,j=0}^{n-1}w_{i,j}Z_iZ_j +\\end{align} +$$ + +where the sum runs over $i