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The complexity of the consistency and Nrepresentability problems for quantum states
"... QMA (Quantum MerlinArthur) is the quantum analogue of the class NP. There are a few QMAcomplete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMAcomplete problems which are very different from those known previo ..."
Abstract

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QMA (Quantum MerlinArthur) is the quantum analogue of the class NP. There are a few QMAcomplete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMAcomplete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is “Consistency of Local Density Matrices”: given a collection of density matrices describing different subsets of an nqubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMAcomplete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, “Fermionic Local Hamiltonian” and “Nrepresentability, ” are QMAcomplete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. Nrepresentability is particularly interesting, as it is a key component