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Abstract: We determine the computational difficulty of finding ground states ofone-dimensional 1D Hamiltonians which are known to be Matrix Product StatesMPS. To this end, we construct a class of 1D frustration free Hamiltonianswith unique MPS ground states and a polynomial gap above, for which finding theground state is at least as hard as factoring. By lifting the requirement of aunique ground state, we obtain a class for which finding the ground statesolves an NP-complete problem. Therefore, for these Hamiltonians it is not evenpossible to certify that the ground state has been found. Our results thusimply that in order to prove convergence of variational methods over MPS, asthe Density Matrix Renormalization Group, one has to put more requirements thanjust MPS ground states and a polynomial spectral gap.



Author: Norbert Schuch, Ignacio Cirac, Frank Verstraete

Source: https://arxiv.org/







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