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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.

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  • $\begingroup$ Hello! I am new here so not yet comfortable with the usage of the platform. There is nothing such as HA. HC is the cost hamiltonian which corresponds to the objective function to be optimized and HB is the operator which is summation of pauli-z operators on ith qubit. HB is the mixing operator. $\endgroup$ – siddhi mali Sep 3 '20 at 16:40
  • $\begingroup$ ah, I see you meant $H_B$ and $H_C$, sorry I thought they were acronyms. Anyway, if you are just asking what a local minimum is in general, the question doesn't have much with quantum mechanics per se. Did you have a look e.g. at the Wiki page? A local minimum is just a minimum for the function that is not the global minimum $\endgroup$ – glS Sep 3 '20 at 20:34
  • $\begingroup$ I am not just looking for local minimum. In the blog, they mentioned, 'the mixing operator helps escaping the local minimum.' What I can think is although I am not sure, that they trying to find some ground sate and that is why trying to reach the global minimum. And how HB is helping achieving that? I want see whole mathematical picture. I am new to QC. $\endgroup$ – siddhi mali Sep 4 '20 at 4:32
  • $\begingroup$ so, are you asking what is meant by "local minimum" here? or what is a local minimum in general? or why they are looking for a local minimum in this particular context? Can you clarify what exactly is the question? $\endgroup$ – glS Sep 4 '20 at 7:34
  • $\begingroup$ I am asking what is meant by "local minimum" here? and why are looking for a local minimum in this particular context? $\endgroup$ – siddhi mali Sep 4 '20 at 7:57
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I want to know what do they mean by local minimum?

In general any optimization problem's goal is to either maximize or minimize the objective function.
The local minimums are the candidate solutions to your optimization problem. Same as in the blog that you've mentioned!

In general, QAOA (Quantum Approximate Optimization Algorithm) tries to solve same optimization problems but with a different approach than classical algorithms.

There is this nice video from D-Wave in which they explain, in an abstract level, how optimization problems could be solved using characteristics (minimal energy state) of quantum systems.

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  • $\begingroup$ Thank you so much! $\endgroup$ – siddhi mali Sep 5 '20 at 4:27

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