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The complexity of antiferromagnetic 2-qubit interactions and 2D lattices

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Estimation of the minimum eigenvalue of a quantum Hamiltonian can be formalised as the Local Hamiltonian problem. In one natural special case of the Local Hamiltonian problem, the same 2-local interaction, with differing weights, is applied across each pair of qubits. I will talk about some recent work classifying the computational complexity of this problem when some additional physically motivated restrictions are made to these weights. In particular we consider the case where these weights are all positive and/or that the interactions are restricted to the edges of a 2D square (or triangular) lattice. For most interactions we are able to classify the complexity as either QMA -complete or contained in StoqMA.

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