Towards sample-optimal learning of bosonic Gaussian quantum states
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Quantum Physics arXiv:2603.18136 (quant-ph) [Submitted on 18 Mar 2026] Title:Towards sample-optimal learning of bosonic Gaussian quantum states Authors:Senrui Chen, Francesco Anna Mele, Marco Fanizza, Alfred Li, Zachary Mann, Hsin-Yuan Huang, Yanbei Chen, John Preskill View a PDF of the paper titled Towards sample-optimal learning of bosonic Gaussian quantum states, by Senrui Chen and 7 other authors View PDF HTML (experimental) Abstract:Continuous-variable systems enable key quantum technologies in computation, communication, and sensing. Bosonic Gaussian states emerge naturally in various such applications, including gravitational-wave and dark-matter detection. A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible. Despite decades-long exploration, the ultimate efficiency limit remains unclear. In this work, we study the necessary and sufficient number of copies to learn an $n$-mode Gaussian state, with energy less than $E$, to $\varepsilon$ trace distance with high probability. We prove a lower bound of $\Omega(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and ${\Omega}(n^2/\varepsilon^2)$ for arbitrary measurements. We further show an upper bound of $\widetilde{O}(n^2/\varepsilon^2)$ given that the Gaussian state is promised to be either pure or passive. Interestingly, while Gaussian measurements suffice for nearly optimal learning of pure Gaussian states, non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. Finally, focusing on learning single-mode Gaussian states via non-entangling Gaussian measurements, we provide a nearly tight bound of $\widetilde\Theta(E/\varepsilon^2)$ for any non-adaptive schemes, showing adaptivity is indispensable for nearly energy-independent scaling. As a byproduct, we establish sharp bounds on the trace distance between Gaussian states in terms of the total variation distance between their Wigner distributions, and obtain a nearly tight sample complexity bound for learning the Wigner distribution of any Gaussian state to $\varepsilon$ total variation distance. Our results greatly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications. Comments: Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Machine Learning (cs.LG); Mathematical Physics (math-ph) Cite as: arXiv:2603.18136 [quant-ph] (or arXiv:2603.18136v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.18136 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Senrui Chen [view email] [v1] Wed, 18 Mar 2026 18:00:00 UTC (629 KB) Full-text links: Access Paper: View a PDF of the paper titled Towards sample-optimal learning of bosonic Gaussian quantum states, by Senrui Chen and 7 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-03 Change to browse by: cs cs.IT cs.LG math math-ph math.IT math.MP References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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