Locally accurate matrix product state approximations with constant bond dimension for ground states of gapped 1D models

The numerical success of the DMRG method has been explained by the observation that it can be recast as a variational algorithm over the set of matrix product states (MPSs) with a specified bond dimension. The bond dimension need only increase like a polynomial in the number of sites on the 1D chain to guarantee that some element of the MPS manifold represents a good approximation to the ground state of a given gapped local Hamiltonian. But DMRG is often successful even for very small values of the bond dimension. We provide a partial justification for this success by showing that the MPS bond dimension may be kept constant as the number of sites increases if one desires an approximation that is good only in a local sense, that is, the reduced density matrix of the true ground state is close to that of the approximating MPS when all but a constant segment of the chain is traced out. While a similar result was known for matrix product operator (MPO) approximations, MPSs are superior to MPOs as an ansatz for variational algorithms since verifying that a certain MPO is positive (and thus represents a valid quantum state) is difficult, whereas MPSs do not have this issue.