{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T02:48:37Z","timestamp":1774925317943,"version":"3.50.1"},"reference-count":60,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T00:00:00Z","timestamp":1710979200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000936","name":"Gordon and Betty Moore Foundation","doi-asserted-by":"crossref","award":["GBMF8686"],"award-info":[{"award-number":["GBMF8686"]}],"id":[{"id":"10.13039\/100000936","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"We study classical shadows protocols based on randomized measurements in n<\/mml:mi><\/mml:math>-qubit entangled bases, generalizing the random Pauli measurement protocol (n<\/mml:mi>=<\/mml:mo>1<\/mml:mn><\/mml:math>). We show that entangled measurements (n<\/mml:mi>&#x2265;<\/mml:mo>2<\/mml:mn><\/mml:math>) enable nontrivial and potentially advantageous trade-offs in the sample complexity of learning Pauli expectation values. This is sharply illustrated by shadows based on two-qubit Bell measurements: the scaling of sample complexity with Pauli weight k<\/mml:mi><\/mml:math> improves quadratically (from &#x223C;<\/mml:mo>3<\/mml:mn>k<\/mml:mi><\/mml:msup><\/mml:math> down to &#x223C;<\/mml:mo>3<\/mml:mn>k<\/mml:mi>\/<\/mml:mo><\/mml:mrow>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>) for many operators, while others become impossible to learn. Tuning the amount of entanglement in the measurement bases defines a family of protocols that interpolate between Pauli and Bell shadows, retaining some of the benefits of both. For large n<\/mml:mi><\/mml:math>, we show that randomized measurements in n<\/mml:mi><\/mml:math>-qubit GHZ bases further improve the best scaling to &#x223C;<\/mml:mo>(<\/mml:mo>3<\/mml:mn>\/<\/mml:mo><\/mml:mrow>2<\/mml:mn>)<\/mml:mo>k<\/mml:mi><\/mml:msup><\/mml:math>, albeit on an increasingly restricted set of operators. Despite their simplicity and lower hardware requirements, these protocols can match or outperform recently-introduced \"shallow shadows\" in some practically-relevant Pauli estimation tasks.<\/jats:p>","DOI":"10.22331\/q-2024-03-21-1293","type":"journal-article","created":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T10:23:01Z","timestamp":1711016581000},"page":"1293","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":16,"title":["Classical shadows based on locally-entangled measurements"],"prefix":"10.22331","volume":"8","author":[{"given":"Matteo","family":"Ippoliti","sequence":"first","affiliation":[{"name":"Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA"},{"name":"Department of Physics, Stanford University, Stanford, CA 94305, USA"}]}],"member":"9598","published-online":{"date-parts":[[2024,3,21]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Predicting many properties of a quantum system from very few measurements''. 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