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I was wondering if one can think of a more general relation between alleviating conditions for the state in which the evolution takes palce in AQC paradigm and constraining the structure of the Hamiltonian.
To illustrate: if we constrain ourselves to a stoquastic Hamiltonian but allow for evolution in an excited state, such setup is as poewrful as original AQC. An example would be from Jordan, Gosset, and Love work https://arxiv.org/abs/0905.4755 mentioned in https://arxiv.org/abs/1611.04471

Our definition of StoqAQC (Definition 6) stipulates that the computation must proceed in the ground state. It turns out that if this condition is relaxed, computation with stoquastic Hamiltonians is as powerful as AQC, i.e., it is universal. Here we review a construction by Jordan, Gosset, and Love (2010) of a 3-local stoquastic Hamiltonian that, by allowing for excited state evolution, is both QMA complete and universal for AQC.

Is there any point to look at such case in a more general way and look for relations between the structure of the Hamiltonian and the type of state/states used in evolution of the system?

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