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Similarly to available classical algorithms for Lasso, our quantum algorithm provides the full regularisation path as the penalty term varies, but quadratically faster per iteration under specific conditions. A quadratic speedup on the number of features <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><\/mml:math> is possible by using the simple quantum minimum-finding subroutine from D\u00fcrr and Hoyer (arXiv'96) in order to obtain the joining time at each iteration. We then improve upon this simple quantum algorithm and obtain a quadratic speedup both in the number of features <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><\/mml:math> and the number of observations <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math> by using the approximate quantum minimum-finding subroutine from Chen and de Wolf (ICALP'23). In order to do so, we approximately compute the joining times to be searched over by the approximate quantum minimum-finding subroutine. As another main contribution, we prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm (and therefore our quantum algorithm) is robust to errors. This means that it still outputs a path that minimises the Lasso cost function up to a small error if the joining times are only approximately computed. Furthermore, we show that, when the observations are sampled from a Gaussian distribution, our quantum algorithm's complexity only depends polylogarithmically on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math>, exponentially better than the classical LARS algorithm, while keeping the quadratic improvement on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><\/mml:math>. Moreover, we propose a dequantised version of our quantum algorithm that also retains the polylogarithmic dependence on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math>, albeit presenting the linear scaling on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><\/mml:math> from the standard LARS algorithm. Finally, we prove query lower bounds for classical and quantum Lasso algorithms.<\/jats:p>","DOI":"10.22331\/q-2025-03-25-1674","type":"journal-article","created":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T19:20:49Z","timestamp":1742930449000},"page":"1674","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Quantum Algorithms for the Pathwise Lasso"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8265-7334","authenticated-orcid":false,"given":"Joao F.","family":"Doriguello","sequence":"first","affiliation":[{"name":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics, Budapest, Hungary"},{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore"}]},{"given":"Debbie","family":"Lim","sequence":"additional","affiliation":[{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore"},{"name":"Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Latvia"}]},{"given":"Chi Seng","family":"Pun","sequence":"additional","affiliation":[{"name":"School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore"}]},{"given":"Patrick","family":"Rebentrost","sequence":"additional","affiliation":[{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore"},{"name":"Department of Computer Science, National University of Singapore, Singapore"}]},{"given":"Tushar","family":"Vaidya","sequence":"additional","affiliation":[{"name":"School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore"}]}],"member":"9598","published-online":{"date-parts":[[2025,3,25]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Jonathan Allcock, Jinge Bao, Joao F. 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