{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:41:01Z","timestamp":1777516861907,"version":"3.51.4"},"reference-count":31,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T00:00:00Z","timestamp":1693267200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The Research Foundation \u2013 Flanders","award":["G006020N"],"award-info":[{"award-number":["G006020N"]}]},{"name":"Excellence of Science (EOS) research programme","award":["G0H1422N"],"award-info":[{"award-number":["G0H1422N"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"Linear optical quantum circuits with photon number resolving (PNR) detectors are used for both Gaussian Boson Sampling (GBS) and for the preparation of non-Gaussian states such as Gottesman-Kitaev-Preskill (GKP), cat and NOON states. They are crucial in many schemes of quantum computing and quantum metrology. Classically optimizing circuits with PNR detectors is challenging due to their exponentially large Hilbert space, and quadratically more challenging in the presence of decoherence as state vectors are replaced by density matrices. To tackle this problem, we introduce a family of algorithms that calculate detection probabilities, conditional states (as well as their gradients with respect to circuit parametrizations) with a complexity that is comparable to the noiseless case. As a consequence we can simulate and optimize circuits with twice the number of modes as we could before, using the same resources. More precisely, for an M<\/mml:mi><\/mml:math>-mode noisy circuit with detected modes D<\/mml:mi><\/mml:math> and undetected modes U<\/mml:mi><\/mml:math>, the complexity of our algorithm is O<\/mml:mi>(<\/mml:mo>M<\/mml:mi>2<\/mml:mn><\/mml:msup>&#x220F;<\/mml:mo>i<\/mml:mi>&#x2208;<\/mml:mo>U<\/mml:mi><\/mml:mrow><\/mml:munder>C<\/mml:mi>i<\/mml:mi>2<\/mml:mn><\/mml:msubsup>&#x220F;<\/mml:mo>i<\/mml:mi>&#x2208;<\/mml:mo>D<\/mml:mi><\/mml:mrow><\/mml:munder>C<\/mml:mi>i<\/mml:mi><\/mml:msub>)<\/mml:mo><\/mml:math>, rather than O<\/mml:mi>(<\/mml:mo>M<\/mml:mi>2<\/mml:mn><\/mml:msup>&#x220F;<\/mml:mo>i<\/mml:mi>&#x2208;<\/mml:mo>D<\/mml:mi>&#x222A;<\/mml:mo>U<\/mml:mi><\/mml:mrow><\/mml:munder>C<\/mml:mi>i<\/mml:mi>2<\/mml:mn><\/mml:msubsup>)<\/mml:mo><\/mml:math>, where C<\/mml:mi>i<\/mml:mi><\/mml:msub><\/mml:math> is the Fock cutoff of mode i<\/mml:mi><\/mml:math>. As a particular case, our approach offers a full quadratic speedup for calculating detection probabilities, as in that case all modes are detected. Finally, these algorithms are implemented and ready to use in the open-source photonic optimization library MrMustard.<\/jats:p>","DOI":"10.22331\/q-2023-08-29-1097","type":"journal-article","created":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T15:19:47Z","timestamp":1693322387000},"page":"1097","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["A Quadratic Speedup in the Optimization of Noisy Quantum Optical Circuits"],"prefix":"10.22331","volume":"7","author":[{"given":"Robbe","family":"De Prins","sequence":"first","affiliation":[{"name":"Photonics Research Group, INTEC, Ghent University \u2013 imec, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuan","family":"Yao","sequence":"additional","affiliation":[{"name":"T\u00e9l\u00e9com Paris and Institut Polytechnique de Paris, LTCI, 20 Place Marguerite Perey, 91120 Palaiseau, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anuj","family":"Apte","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"},{"name":"Kadanoff Center for Theoretical Physics & Enrico Fermi Institute, Department of Physics, University of Chicago, Chicago, IL 60637"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Filippo M.","family":"Miatto","sequence":"additional","affiliation":[{"name":"T\u00e9l\u00e9com Paris and Institut Polytechnique de Paris, LTCI, 20 Place Marguerite Perey, 91120 Palaiseau, France"},{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2023,8,29]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Juan Miguel Arrazola and Thomas R. 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Tensor network algorithm for simulating experimental Gaussian boson sampling. arXiv preprint arXiv:2306.03709, 2023. 10.48550\/arXiv.2306.03709.","DOI":"10.48550\/arXiv.2306.03709"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Nicol\u00e1s Quesada. Franck-Condon factors by counting perfect matchings of graphs with loops. The Journal of chemical physics, 150 (16): 164113, 2019. 10.1063\/1.5086387.","DOI":"10.1063\/1.5086387"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Nicol\u00e1s Quesada, Luke G. Helt, Josh Izaac, Juan Miguel Arrazola, Reihaneh Shahrokhshahi, Casey R. Myers, and Krishna K. Sabapathy. Simulating realistic non-Gaussian state preparation. Phys. Rev. A, 100: 022341, August 2019. 10.1103\/PhysRevA.100.022341.","DOI":"10.1103\/PhysRevA.100.022341"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Krishna K. Sabapathy, Haoyu Qi, Josh Izaac, and Christian Weedbrook. 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