Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians

A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length N of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can he improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length-k segments of the chain, the reduced density matrices of our approximations are epsilon-close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like (k/epsilon)(1+o(1)), and at the expense of worse but still poly(k, 1/epsilon) scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension exp(O(k/epsilon)), which is exponentially worse, but still independent of N. Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find O(1)-accurate local approximations to the ground state in T(N) time implies the ability to estimate the ground state energy to O(1) precision in O(T(N) log(N )) time.