{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T15:34:48Z","timestamp":1774539288491,"version":"3.50.1"},"reference-count":61,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T00:00:00Z","timestamp":1728518400000},"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":"There has been much recent interest in near-term applications of quantum computers, i.e., using quantum circuits that have short decoherence times due to hardware limitations. Variational quantum algorithms (VQA), wherein an optimization algorithm implemented on a classical computer evaluates a parametrized quantum circuit as an objective function, are a leading framework in this space. An enormous breadth of algorithms in this framework have been proposed for solving a range of problems in machine learning, forecasting, applied physics, and combinatorial optimization, among others.In this paper, we analyze the iteration complexity of VQA, that is, the number of steps that VQA requires until its iterates satisfy a surrogate measure of optimality. We argue that although VQA procedures incorporate algorithms that can, in the idealized case, be modeled as classic procedures in the optimization literature, the particular nature of noise in near-term devices invalidates the claim of applicability of off-the-shelf analyses of these algorithms. Specifically, noise makes the evaluations of the objective function via quantum circuits b<\/mml:mi>i<\/mml:mi>a<\/mml:mi>s<\/mml:mi>e<\/mml:mi>d<\/mml:mi><\/mml:math>. Commonly used optimization procedures, such as SPSA and the parameter shift rule, can thus be seen as derivative-free optimization algorithms with biased function evaluations, for which there are currently no iteration complexity guarantees in the literature. We derive the missing guarantees and find that the rate of convergence is unaffected. However, the level of bias contributes unfavorably to both the constant therein, and the asymptotic distance to stationarity, i.e., the more bias, the farther one is guaranteed, at best, to reach a stationary point of the VQA objective.<\/jats:p>","DOI":"10.22331\/q-2024-10-10-1495","type":"journal-article","created":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T09:17:17Z","timestamp":1728551837000},"page":"1495","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Iteration Complexity of Variational Quantum Algorithms"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2229-8824","authenticated-orcid":false,"given":"Vyacheslav","family":"Kungurtsev","sequence":"first","affiliation":[{"name":"Department of Computer Science, Czech Technical University in Prague, Karlovo nam. 13, Prague 2, Czech Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3850-4979","authenticated-orcid":false,"given":"Georgios","family":"Korpas","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Czech Technical University in Prague, Karlovo nam. 13, Prague 2, Czech Republic"},{"name":"Archimedes Research Unit on AI, Data Science and Algorithms, Athena Research and Innovation Center, 15125 Marousi, Greece"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0839-0691","authenticated-orcid":false,"given":"Jakub","family":"Marecek","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Czech Technical University in Prague, Karlovo nam. 13, Prague 2, Czech Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4497-2093","authenticated-orcid":false,"given":"Elton Yechao","family":"Zhu","sequence":"additional","affiliation":[{"name":"Fidelity Center for Applied Technology, FMR LLC, Boston, MA 02210, USA"}]}],"member":"9598","published-online":{"date-parts":[[2024,10,10]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Al\u00e1 n Aspuru-Guzik. ``The theory of variational hybrid quantum-classical algorithms''. 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