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Compared to other observable estimation protocols based on classical shadows and measurement frames, our approach offers several advantages: (i) it can be optimised to provide lower statistical error, resulting in a reduced measurement budget to achieve a specified estimation precision; (ii) it scales to a large number of qubits due to the tensor network structure; (iii) it can be applied to any measurement protocol with measurement operators that have an efficient tensor-network representation. We benchmark the method through various numerical examples, including spin and chemical systems, and show that our method can provide statistical error that are orders of magnitude lower than the ones given by classical shadows.<\/jats:p>","DOI":"10.22331\/q-2025-07-23-1812","type":"journal-article","created":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T14:40:38Z","timestamp":1753281638000},"page":"1812","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Low variance estimations of many observables with tensor networks and informationally-complete measurements"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0056-0660","authenticated-orcid":false,"given":"Stefano","family":"Mangini","sequence":"first","affiliation":[{"name":"Algorithmiq Ltd, Kanavakatu 3C 00160 Helsinki, Finland."},{"name":"QTF Centre of Excellence, Department of Physics, University of Helsinki, P.O. Box 43, FI-00014 Helsinki, Finland."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2704-3049","authenticated-orcid":false,"given":"Daniel","family":"Cavalcanti","sequence":"additional","affiliation":[{"name":"Algorithmiq Ltd, Kanavakatu 3C 00160 Helsinki, Finland."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,7,23]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Marcus Cramer, Martin B. Plenio, Steven T. Flammia, Rolando Somma, David Gross, Stephen D. Bartlett, Olivier Landon-Cardinal, David Poulin, and Yi-Kai Liu. ``Efficient quantum state tomography''. Nature Communications 1 (2010).","DOI":"10.1038\/ncomms1147"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Juan Carrasquilla, Giacomo Torlai, Roger G. Melko, and Leandro Aolita. ``Reconstructing quantum states with generative models''. 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