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In QAOA and AQC the problem Hamiltonian is always a Phase Hamiltonian (meaning only phases are added)

Is that part of the QAOA and AQC definition or it is only used because it is convenient and work?

If one uses non phase Hamiltonians with QAOA or AQC, (theoretically the Adiabatic theorem would still work) would it still be considered to be QAOA and AQC?

Thanks for reading.

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