Metric Self-Dual Completion and Optimal Additive Hardness for Quantum and Graph-State Distance

Quantum Physics arXiv:2609.22669 (quant-ph) [Submitted on 19 Sep 2026] Title:Metric Self-Dual Completion and Optimal Additive Hardness for Quantum and Graph-State Distance Authors:Rafail Ostrovsky View a PDF of the paper titled Metric Self-Dual Completion and Optimal Additive Hardness for Quantum and Graph-State Distance, by Rafail Ostrovsky View PDF HTML (experimental) Abstract:We prove that the quantum code distance is NP-hard to approximate within an additive error of $c N$, for some constant $c >0$, where $N$ is the number of qubits. Our reductions are deterministic. This improves the previous square-root additive gap to $\Omega(N)$ and resolves the explicitly stated linear-gap question of Kapshikar and Kundu. Our result holds for CSS codes with identical $X$- and $Z$-check spaces, and with a constant rate and constant relative distance. For every fixed $\lambda>1$, there is a constant $c>0$ such that hardness still holds even when every nonidentity stabilizer has weight greater than $\lambda$ times the quantum distance. We also improve the hardness gap of graph state distance on $N$ vertices of Grigorescu, Jha, and Samperton from cube-root to $\Omega(N)$, resolving their explicitly stated open question. Both hardness results are asymptotically optimal since both distances are at most $N$. Our graph state distance hardness result holds for balanced bipartite graphs with a binary adjacency matrix that is its own inverse (mod 2). Our main technique for both hardness bounds above is classical: we show how to convert any code $C$ of length $m$ into a self-dual code $A(C)$ of length $N=\Theta(m)$ while exactly doubling the original coset metric. The conversion is deterministic and efficient. We call it the metric self-dual completion of $C$. It comes with a linear embedding $\tau: \mathbb F_2^m \hookrightarrow \mathbb F_2^N$. The embedding doubles all Hamming distances between vectors in $\mathbb F_2^m$ and all pairwise distances between corresponding cosets. The embedding also guarantees that all codewords of $A(C)$ of weight at most $2m$ are exactly $\tau(C)$. Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC); Information Theory (cs.IT) MSC classes: 94B05 (Primary) 68Q17, 81P70 (Secondary) ACM classes: E.4; F.1.3 Cite as: arXiv:2609.22669 [quant-ph] (or arXiv:2609.22669v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2609.22669 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Rafail Ostrovsky [view email] [v1] Sat, 19 Sep 2026 00:57:30 UTC (20 KB) Full-text links: Access Paper: View a PDF of the paper titled Metric Self-Dual Completion and Optimal Additive Hardness for Quantum and Graph-State Distance, by Rafail OstrovskyView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-09 Change to browse by: cs cs.CC cs.IT math math.IT 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?)