Optimal learning of quantum channels in diamond distance

Quantum Physics arXiv:2512.10214 (quant-ph) [Submitted on 11 Dec 2025] Title:Optimal learning of quantum channels in diamond distance Authors:Antonio Anna Mele, Lennart Bittel View a PDF of the paper titled Optimal learning of quantum channels in diamond distance, by Antonio Anna Mele and 1 other authors View PDF HTML (experimental) Abstract:Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory and a key primitive for characterising noisy quantum devices. A long-standing open question is to determine the optimal number of uses of an unknown channel required to learn it in diamond distance, the standard measure of worst-case distinguishability between quantum processes. Here we show that a quantum channel acting on a $d$-dimensional system can be estimated to accuracy $\varepsilon$ in diamond distance using $O(d^4/\varepsilon^2)$ channel uses. This scaling is essentially optimal, as it matches lower bounds up to logarithmic factors. Our analysis extends to channels with input and output dimensions $d_{\mathrm{in}}$ and $d_{\mathrm{out}}$ and Kraus rank at most $k$, for which $O(d_{\mathrm{in}} d_{\mathrm{out}} k/\varepsilon^2)$ channel uses suffice, interpolating between unitary and fully generic channels. As by-products, we obtain, to the best of our knowledge, the first essentially optimal strategies for operator-norm learning of binary POVMs and isometries, and we recover optimal trace-distance tomography for fixed-rank states. Our approach consists of using the channel only non-adaptively to prepare copies of the Choi state, purify them in parallel, perform sample-optimal pure-state tomography on the purifications, and analyse the resulting estimator directly in diamond distance via its semidefinite-program characterisation. While the sample complexity of state tomography in trace distance is by now well understood, our results finally settle the corresponding problem for quantum channels in diamond distance. Comments: Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS) Cite as: arXiv:2512.10214 [quant-ph] (or arXiv:2512.10214v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.10214 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Antonio Anna Mele [view email] [v1] Thu, 11 Dec 2025 02:04:03 UTC (328 KB) Full-text links: Access Paper: View a PDF of the paper titled Optimal learning of quantum channels in diamond distance, by Antonio Anna Mele and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-12 Change to browse by: cs cs.CC cs.DS 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?) Links to Code Toggle Papers with Code (What is Papers with Code?) 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?)