Error estimates for extrapolations with matrix-product states

We introduce an error measure for matrix-product states without requiring the relatively costly two-site density-matrix renormalization group (2DMRG). This error measure is based on an approximation of the full variance <psi broken vertical bar(H) over cap - E)(2) broken vertical bar psi >. When applied to a series of matrix-product states at different bond dimensions obtained from a single-site density-matrix renormalization group (1DMRG) calculation, it allows for the extrapolation of observables towards the zero-error case representing the exact ground state of the system. The calculation of the error measure is split into a sequential part of cost equivalent to two calculations of <psi broken vertical bar(H) over cap broken vertical bar psi > and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of this error measure is demonstrated by four examples: the L = 30, S = 1/2 Heisenberg chain, the L = 50 Hubbard chain, an electronic model with long-range Coulomb-like interactions, and the Hubbard model on a cylinder with a size of 10 x 4. Extrapolation in this error measure is shown to be on par with extrapolation in the 2DMRG truncation error or the full variance <psi broken vertical bar(H) over cap - E)(2) broken vertical bar psi > at a fraction of the computational effort.