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Despite its importance, a similar backpropagation-like scaling for gradient evaluation of parameterised quantum circuits has remained elusive. Currently, the most popular method requires sampling from a number of circuits that scales with the number of circuit parameters, making training of large-scale quantum circuits prohibitively expensive in practice. Here we address this problem by introducing a class of structured circuits that are not known to be classically simulable and admit gradient estimation with significantly fewer circuits. In the simplest case \u2013 for which the parameters feed into commuting quantum gates \u2013 these circuits allow for fast estimation of the gradient, higher order partial derivatives and the Fisher information matrix. Moreover, specific families of parameterised circuits exist for which the scaling of gradient estimation is in line with classical backpropagation, and can thus be trained at scale. In a toy classification problem on 16 qubits, such circuits show competitive performance with other methods, while reducing the training cost by about two orders of magnitude.<\/jats:p>","DOI":"10.22331\/q-2025-10-02-1873","type":"journal-article","created":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T13:02:13Z","timestamp":1759410133000},"page":"1873","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Backpropagation scaling in parameterised quantum circuits"],"prefix":"10.22331","volume":"9","author":[{"given":"Joseph","family":"Bowles","sequence":"first","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"}]},{"given":"David","family":"Wierichs","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"}]},{"given":"Chae-Yeun","family":"Park","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"}]}],"member":"9598","published-online":{"date-parts":[[2025,10,2]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. ``Learning representations by back-propagating errors''. 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