{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T17:12:55Z","timestamp":1780765975323,"version":"3.54.1"},"reference-count":60,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2020,8,31]],"date-time":"2020-08-31T00:00:00Z","timestamp":1598832000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>s<\/mml:mi><mml:mi>t<\/mml:mi><mml:mi>o<\/mml:mi><mml:mi>c<\/mml:mi><mml:mi>h<\/mml:mi><mml:mi>a<\/mml:mi><mml:mi>s<\/mml:mi><mml:mi>t<\/mml:mi><mml:mi>i<\/mml:mi><mml:mi>c<\/mml:mi><\/mml:math>gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math>measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math>. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math>-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain ``doubly stochastic'' gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.<\/jats:p>","DOI":"10.22331\/q-2020-08-31-314","type":"journal-article","created":{"date-parts":[[2020,8,31]],"date-time":"2020-08-31T13:39:38Z","timestamp":1598881178000},"page":"314","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":210,"title":["Stochastic gradient descent for hybrid quantum-classical optimization"],"prefix":"10.22331","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6202-8864","authenticated-orcid":false,"given":"Ryan","family":"Sweke","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6224-1964","authenticated-orcid":false,"given":"Frederik","family":"Wilde","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1533-8015","authenticated-orcid":false,"given":"Johannes","family":"Meyer","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8626-168X","authenticated-orcid":false,"given":"Maria","family":"Schuld","sequence":"additional","affiliation":[{"name":"Xanadu, 777 Bay Street, Toronto, Ontario, Canada"},{"name":"Quantum Research Group, University of KwaZulu-Natal, 4000 Durban, South Africa"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8706-1732","authenticated-orcid":false,"given":"Paul K.","family":"Faehrmann","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Barth\u00e9l\u00e9my","family":"Meynard-Piganeau","sequence":"additional","affiliation":[{"name":"Department of Physics, Ecole Polytechnique, Palaiseau, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3033-1292","authenticated-orcid":false,"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"},{"name":"Helmholtz Center Berlin, 14109 Berlin, Germany"},{"name":"Department of Mathematics and Computer Science, Freie Universit\u00e4t Berlin, D-14195 Berlin"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2020,8,31]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"J. 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