I am trying to understand what exactly happens in QAOA. I am reading this blog which says,
We would be just repetitively applying $U_C$. But once we got into a state which is the eigenstate of $H_C$ we wouldn’t get any further. This is basic linear algebra — if we apply an operator to its eigenvector, it can change its length, but not direction. The same applies if we had a $H_B$ which commutes with $H_C$. So we need this intermediate step of applying $H_B$ which allows us to escape from the local minimum. How do we make sure we escape it? Well, that’s where the classical optimization loop is useful – we try to find the right values of the parameters $β$ and $γ$ which make it happen.
I want to know what do they mean by local minimum?
It also says,
It doesn’t need to be just $\sum_i^N \sigma_i^x$. We want it to be something that does not commute with $H_C$ and this choice of $H_B$ meets this requirement and is super easy to implement.
What does that mean?
$H_C$ is the cost hamiltonian which corresponds to the objective function to be optimized and $H_B$ is the operator which is summation of Pauli-z operators on $i$-th qubit. $H_B$ is the mixing operator.