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A central result in the study of Quantum Hamiltonian Complexity is that thek-Local hamiltonian problem is QMA-complete. In that problem, we must decide ifthe lowest eigenvalue of a Hamiltonian is bounded below some value, or aboveanother, promised one of these is true. Given the ground state of theHamiltonian, a quantum computer can determine this question, even if the groundstate itself may not be efficiently quantum preparable. Kitaev's proof ofQMA-completeness encodes a unitary quantum circuit in QMA into the ground spaceof a Hamiltonian. However, we now have quantum computing models based onmeasurement instead of unitary evolution, furthermore we can use post-selectedmeasurement as an additional computational tool. In this work, we generaliseKitaev's construction to allow for non-unitary evolution includingpost-selection. Furthermore, we consider a type of post-selection under whichthe construction is consistent, which we call tame post-selection. We considerthe computational complexity consequences of this construction and thenconsider how the probability of an event upon which we are post-selectingaffects the gap between the ground state energy and the energy of the firstexcited state of its corresponding Hamiltonian. We provide numerical evidencethat the two are not immediately related, by giving a family of circuits wherethe probability of an event upon which we post-select is exponentially small,but the gap in the energy levels of the Hamiltonian decreases as a polynomial.

Non-Unitary Quantum Computation in the Ground Space of Local Hamiltonians