{"id":197258,"date":"2024-10-08T21:22:29","date_gmt":"2024-10-09T02:22:29","guid":{"rendered":"https:\/\/lifeboat.com\/blog\/2024\/10\/quantum-state-tomography-with-locally-purified-density-operators-and-local-measurements"},"modified":"2024-10-08T21:22:29","modified_gmt":"2024-10-09T02:22:29","slug":"quantum-state-tomography-with-locally-purified-density-operators-and-local-measurements","status":"publish","type":"post","link":"https:\/\/lifeboat.com\/blog\/2024\/10\/quantum-state-tomography-with-locally-purified-density-operators-and-local-measurements","title":{"rendered":"Quantum state tomography with locally purified density operators and local measurements"},"content":{"rendered":"<p><a class=\"aligncenter blog-photo\" href=\"https:\/\/lifeboat.com\/blog.images\/quantum-state-tomography-with-locally-purified-density-operators-and-local-measurements.jpg\"><\/a><\/p>\n<p><i>Quantum<\/i> state tomography plays a fundamental role in characterizing and evaluating the quality of quantum states produced by quantum devices. It serves as a crucial element in the advancement of quantum hardware and software, regardless of the underlying physical implementation and potential applications<sup><a data-track=\"click\" data-track-action=\"reference anchor\" data-track-label=\"link\" data-test=\"citation-ref\" title=\"Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2009).\" href=\"https:\/\/www.nature.com\/articles\/s42005-024-01813-4#ref-CR1\" id=\"ref-link-section-d63626095e362\">1<\/a>,<a data-track=\"click\" data-track-action=\"reference anchor\" data-track-label=\"link\" data-test=\"citation-ref\" title=\"Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).\" href=\"https:\/\/www.nature.com\/articles\/s42005-024-01813-4#ref-CR2\" id=\"ref-link-section-d63626095e362_1\">2<\/a>,<a data-track=\"click\" data-track-action=\"reference anchor\" data-track-label=\"link\" data-test=\"citation-ref\" aria-label=\"Reference 3\" title=\"Gebhart, V. et al. Learning quantum systems. Nat. Rev. Phys. 5141&ndash;156 (2023).\" href=\"https:\/\/www.nature.com\/articles\/s42005-024-01813-4#ref-CR3\" id=\"ref-link-section-d63626095e365\">3<\/a><\/sup>. However, reconstructing the full quantum state becomes prohibitively expensive for large-scale quantum systems that exhibit potential quantum advantages<sup><a data-track=\"click\" data-track-action=\"reference anchor\" data-track-label=\"link\" data-test=\"citation-ref\" aria-label=\"Reference 4\" title=\"Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574505&ndash;510 (2019).\" href=\"https:\/\/www.nature.com\/articles\/s42005-024-01813-4#ref-CR4\" id=\"ref-link-section-d63626095e369\">4<\/a>,<a data-track=\"click\" data-track-action=\"reference anchor\" data-track-label=\"link\" data-test=\"citation-ref\" aria-label=\"Reference 5\" title=\"Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618500&ndash;505 (2023).\" href=\"https:\/\/www.nature.com\/articles\/s42005-024-01813-4#ref-CR5\" id=\"ref-link-section-d63626095e372\">5<\/a><\/sup>, as the number of measurements required increases exponentially with system size.<\/p>\n<p>Recent protocols try to solve this challenge through two main steps: efficient parameterization of quantum states and utilization of carefully designed measurement schemes and classical data postprocessing algorithms. For one-dimensional (1D) systems with area law entanglement, the matrix product state (MPS)<sup>6,7,8,9,10,11,12<\/sup> provides a compressed representation. It requires only a polynomial number of parameters that can be determined from local or global measurement results. Two iterative algorithms using local measurements, singular value thresholding (SVT)<sup>13<\/sup> and maximum likelihood (ML)<sup>14<\/sup>, have been demonstrated in trapped-ion quantum simulators with up to 14 qubits<sup>15<\/sup>. However, SVT is limited to pure states and thus impractical for noisy intermediate-scale quantum (NISQ) systems. Meanwhile, although ML can handle mixed states represented as matrix product operators (MPOs)<sup>16,17<\/sup>, it suffers from inefficient classical data postprocessing.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Quantum state tomography plays a fundamental role in characterizing and evaluating the quality of quantum states produced by quantum devices. It serves as a crucial element in the advancement of quantum hardware and software, regardless of the underlying physical implementation and potential applications1,2,3. However, reconstructing the full quantum state becomes prohibitively expensive for large-scale quantum [\u2026]<\/p>\n","protected":false},"author":511,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[41,1617],"tags":[],"class_list":["post-197258","post","type-post","status-publish","format-standard","hentry","category-information-science","category-quantum-physics"],"_links":{"self":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/197258","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/users\/511"}],"replies":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/comments?post=197258"}],"version-history":[{"count":0,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/197258\/revisions"}],"wp:attachment":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/media?parent=197258"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/categories?post=197258"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/tags?post=197258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}