Zhen Qin, Joseph M. Lukens, Brian T. Kirby, Zhihui Zhu
{"title":"Enhancing quantum state reconstruction with structured classical shadows","authors":"Zhen Qin, Joseph M. Lukens, Brian T. Kirby, Zhihui Zhu","doi":"10.1038/s41534-025-01101-1","DOIUrl":null,"url":null,"abstract":"<p>While classical shadows can efficiently predict key quantum state properties, their suitability for certified quantum state tomography remains uncertain. In this paper, we address this challenge by introducing a projected classical shadow (PCS) that extends the standard classical shadow by incorporating a projection step onto the target subspace. For a general quantum state consisting of <i>n</i> qubits, our method requires a minimum of <i>O</i>(4<sup><i>n</i></sup>) total state copies to achieve a bounded recovery error in the Frobenius norm between the reconstructed and true density matrices, reducing to <i>O</i>(2<sup><i>n</i></sup><i>r</i>) for states of rank <i>r</i> < 2<sup><i>n</i></sup>—meeting information-theoretic optimal bounds in both cases. For matrix product operator states, we demonstrate that the PCS can recover the ground-truth state with <i>O</i>(<i>n</i><sup>2</sup>) total state copies, improving upon the previously established Haar-random bound of <i>O</i>(<i>n</i><sup>3</sup>). Numerical simulations validate our scaling results and demonstrate the practical accuracy of the proposed PCS method.</p>","PeriodicalId":19212,"journal":{"name":"npj Quantum Information","volume":"11 1","pages":""},"PeriodicalIF":8.3000,"publicationDate":"2025-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"npj Quantum Information","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1038/s41534-025-01101-1","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
While classical shadows can efficiently predict key quantum state properties, their suitability for certified quantum state tomography remains uncertain. In this paper, we address this challenge by introducing a projected classical shadow (PCS) that extends the standard classical shadow by incorporating a projection step onto the target subspace. For a general quantum state consisting of n qubits, our method requires a minimum of O(4n) total state copies to achieve a bounded recovery error in the Frobenius norm between the reconstructed and true density matrices, reducing to O(2nr) for states of rank r < 2n—meeting information-theoretic optimal bounds in both cases. For matrix product operator states, we demonstrate that the PCS can recover the ground-truth state with O(n2) total state copies, improving upon the previously established Haar-random bound of O(n3). Numerical simulations validate our scaling results and demonstrate the practical accuracy of the proposed PCS method.
期刊介绍:
The scope of npj Quantum Information spans across all relevant disciplines, fields, approaches and levels and so considers outstanding work ranging from fundamental research to applications and technologies.