Error estimates for extrapolations with matrix-product states

We introduce a new 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 $\langle \psi | ( \hat H - E )^2 |\psi \rangle$. 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 $\langle \psi | \hat H | \psi \rangle$ and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of the new error measure is demonstrated at four examples: the $L=30, S=\frac{1}{2}$ Heisenberg chain, the $L=50$ Hubbard chain, an electronic model with long-range Coulomb-like interactions and the Hubbard model on a cylinder of size $10 \times 4$. Extrapolation in the new error measure is shown to be on-par with extrapolation in the 2DMRG truncation error or the full variance $\langle \psi | ( \hat H - E )^2 |\psi \rangle$ at a fraction of the computational effort.