{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T09:50:12Z","timestamp":1778493012996,"version":"3.51.4"},"reference-count":56,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,2]],"date-time":"2024-02-02T00:00:00Z","timestamp":1706832000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Suzhou Innovation and Entrepreneurship Pioneer Program","award":["ZXL2022424"],"award-info":[{"award-number":["ZXL2022424"]}]},{"name":"Suzhou Innovation and Entrepreneurship Pioneer Program","award":["BE202311"],"award-info":[{"award-number":["BE202311"]}]},{"name":"Jiangsu Province Industrial Foresight and Key Core Technology Research Plan","award":["ZXL2022424"],"award-info":[{"award-number":["ZXL2022424"]}]},{"name":"Jiangsu Province Industrial Foresight and Key Core Technology Research Plan","award":["BE202311"],"award-info":[{"award-number":["BE202311"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The engineering of quantum computers requires the reliable characterization of qubits, quantum operations, and even the entire hardware. Quantum tomography is an indispensable framework in quantum characterization, verification, and validation (QCVV), which has been widely accepted by researchers. According to the tomographic target, quantum tomography can be categorized into quantum state tomography (QST), quantum process tomography (QPT), gate set tomography (GST), process tensor tomography (PTT), and instrument set tomography (IST). Standard quantum tomography toolkits generally consist of basic linear inverse methods and statistical maximum likelihood estimation (MLE)-based methods. Furthermore, the performance of standard methods, including effectiveness and efficiency, has been further developed by exploiting Bayesian estimation, neural networks, matrix completion techniques, etc. In this review, we introduce the fundamental quantum tomography techniques, including QST, QPT, GST, PTT, and IST. We first introduce the details of basic linear inverse methods. Then, the framework of MLE methods with constraints is summarized. Finally, we briefly introduce recent further research in developing the performance of tomography, utilizing some symmetry properties of the target. This review provides a primary getting-start in developing quantum tomography, which promotes quantum computer development.<\/jats:p>","DOI":"10.3390\/sym16020180","type":"journal-article","created":{"date-parts":[[2024,2,2]],"date-time":"2024-02-02T09:24:08Z","timestamp":1706865848000},"page":"180","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Quantum Tomography: From Markovianity to Non-Markovianity"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3962-5772","authenticated-orcid":false,"given":"Tian","family":"Luan","sequence":"first","affiliation":[{"name":"Yangtze Delta Region Industrial Innovation Center of Quantum Science and Technology, Suzhou 215000, China"},{"name":"Quantum Information Center, Southeast University, Nanjing 210096, China"},{"name":"China Academy of Electronics and Information Technology, Beijing 100041, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zetong","family":"Li","sequence":"additional","affiliation":[{"name":"Quantum Information Center, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Congcong","family":"Zheng","sequence":"additional","affiliation":[{"name":"Quantum Information Center, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xueheng","family":"Kuang","sequence":"additional","affiliation":[{"name":"Yangtze Delta Region Industrial Innovation Center of Quantum Science and Technology, Suzhou 215000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xutao","family":"Yu","sequence":"additional","affiliation":[{"name":"Quantum Information Center, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zaichen","family":"Zhang","sequence":"additional","affiliation":[{"name":"Quantum Information Center, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"010201","DOI":"10.1103\/PRXQuantum.2.010201","article-title":"Theory of Quantum System Certification","volume":"2","author":"Kliesch","year":"2021","journal-title":"PRX Quantum"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1038\/s42254-020-0186-4","article-title":"Quantum Certification and Benchmarking","volume":"2","author":"Eisert","year":"2020","journal-title":"Nat. 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