{"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/efficient-algorithms-for-maximum-likelihood","title":"Efficient Algorithms for Maximum Likelihood Decoding in the Surface Code","arxiv_id":"1405.4883","date":"2014-05-19","proceeding":null,"authors":["Sergey Bravyi","Martin Suchara","Alexander Vargo"],"abstract":"We describe two implementations of the optimal error correction algorithm known as the maximum likelihood decoder (MLD) for the 2D surface code with a noiseless syndrome extraction. First, we show how to implement MLD exactly in time $O(n^2)$, where $n$ is the number of code qubits. Our implementation uses a reduction from MLD to simulation of matchgate quantum circuits. This reduction however requires a special noise model with independent bit-flip and phase-flip errors. Secondly, we show how to implement MLD approximately for more general noise models using matrix product states (MPS). Our implementation has running time $O(n\\chi^3)$ where $\\chi$ is a parameter that controls the approximation precision. The key step of our algorithm, borrowed from the DMRG method, is a subroutine for contracting a tensor network on the two-dimensional grid. The subroutine uses MPS with a bond dimension $\\chi$ to approximate the sequence of tensors arising in the course of contraction. We benchmark the MPS-based decoder against the standard minimum weight matching decoder observing a significant reduction of the logical error probability for $\\chi\\ge 4$.","url_abs":"https://arxiv.org/abs/1405.4883v1","url_pdf":"https://arxiv.org/pdf/1405.4883v1.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":"efficient-algorithms-for-maximum-likelihood","repo_url":"https://github.com/amazon-science/latticealgorithms.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"Apache-2.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1405.4883","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}