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I've come across several of the QAOA tutorials, and yet I'm afraid I haven't found any explanation on this. According to the Qiskit tutorial on QAOA, the cost function for the target problem has a series of the real-valued weights $W_{(Q,\bar{Q})}$, which is still present as it is in the problem Hamiltonian $H$ derived from the cost function.

My question is, how can one actually put these real-valued numbers into the Hamiltonian? As far as I understand they represent the energy of the system altogether, but when it comes to implementing QAOA on a real quantum computer I wonder how you can manage to encode those real values as energy in the Hamiltonian.

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The real-valued weights W from the original problem are not encoded for the quantum computer to evaluate. Instead a classical routine applies them once this classical routine has gained access to the expectations that were indeed obtained from measurement samples of a real quantum computer.

The Qiskit documentation actually has some description of this in the Appendix: Evaluation on a quantum computer.

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