Hardness of approximation for minimum-weight decoding of two-dimensional topological quantum codes

Quantum Physics arXiv:2608.17109 (quant-ph) [Submitted on 17 Aug 2026] Title:Hardness of approximation for minimum-weight decoding of two-dimensional topological quantum codes Authors:Louay Bazzi, Georges Khater View a PDF of the paper titled Hardness of approximation for minimum-weight decoding of two-dimensional topological quantum codes, by Louay Bazzi and Georges Khater View PDF HTML (experimental) Abstract:Efficient decoding is essential for the practical realization of fault-tolerant quantum computers. We study the computational complexity of minimum-weight decoding for topological quantum codes. For surface codes under the depolarizing channel, we consider Minimum-Weight decoding, which seeks a minimum-weight Pauli error consistent with both the $X$- and $Z$-syndromes. For color codes under independent $X$- and $Z$-error models, we consider Separate Minimum-Weight decoding. Assuming $P\neq NP$, we establish polynomial additive inapproximability gaps for these problems. Specifically, for the toric code and the $4.8.8$ color code on the torus, no polynomial-time algorithm can always produce a solution whose weight is within $\Omega(N^{1/14})$ of the optimum, where $N$ is the number of qubits. For the planar surface code, we obtain an $\Omega(N^{1/18})$ gap. Our inapproximability results use Håstad's hardness of approximation for MAX-3SAT. Our reduction develops a general, modular framework for embedding logical constraints into coupled primal--dual join problems on a lattice. A key ingredient is a localization argument that controls unintended interactions between different parts of the construction. Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC) Cite as: arXiv:2608.17109 [quant-ph] (or arXiv:2608.17109v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2608.17109 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Louay Bazzi [view email] [v1] Mon, 17 Aug 2026 20:37:08 UTC (3,049 KB) Full-text links: Access Paper: View a PDF of the paper titled Hardness of approximation for minimum-weight decoding of two-dimensional topological quantum codes, by Louay Bazzi and Georges KhaterView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-08 Change to browse by: cs cs.CC References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)