{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/error-estimates-for-extrapolations-with","title":"Error estimates for extrapolations with matrix-product states","arxiv_id":"1711.01104","date":"2017-11-03","proceeding":null,"authors":["C. Hubig","J. Haegeman","U. Schollwöck"],"abstract":"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.","url_abs":"https://arxiv.org/abs/1711.01104v2","url_pdf":"https://arxiv.org/pdf/1711.01104v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"error-estimates-for-extrapolations-with","repo_url":"https://github.com/giacomogiudice/uMPS","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}