{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:30:30Z","timestamp":1787340630969,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"name":"Latvian State Research Programme","award":["NexIT project 1"],"award-info":[{"award-number":["NexIT project 1"]}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["Advanced Grant MQC"],"award-info":[{"award-number":["Advanced Grant MQC"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100011102","name":"Seventh Framework Programme","doi-asserted-by":"publisher","award":["600700"],"award-info":[{"award-number":["600700"]}],"id":[{"id":"10.13039\/100011102","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000005","name":"U.S. Department of Defense","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000005","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs $\\Omega(\\sqrt{N}\/\\log N)$ queries (improving an $\\Omega(N^{1\/4})$ lower bound of Aaronson). Conversely, we show that this 1 versus $\\widetilde{\\Omega}(\\sqrt{N})$ separation is optimal: indeed, any $t$-query quantum algorithm whatsoever can be simulated by an $O(N^{1-1\/2t})$-query randomized algorithm. Thus, resolving an open question of Buhrman et al. [ SIAM J. Comput., 37 (2008), pp. 1387--1400] from 2002, there is no partial Boolean function whose quantum query complexity is constant and whose randomized query complexity is linear. We conjecture that a natural generalization of Forrelation achieves the optimal $t$ versus $\\Omega(N^{1-1\/2t})$ separation for all $t$. As a bonus, we show that this generalization is ${BQP}$-complete. This yields what is arguably the simplest ${BQP}$-complete problem yet known and gives a second sense in which Forrelation \u201ccaptures the maximum power of quantum computation.\u201d<\/jats:p>","DOI":"10.1137\/15m1050902","type":"journal-article","created":{"date-parts":[[2018,6,27]],"date-time":"2018-06-27T15:57:32Z","timestamp":1530115052000},"page":"982-1038","source":"Crossref","is-referenced-by-count":34,"title":["Forrelation: A Problem That Optimally Separates Quantum from Classical Computing"],"prefix":"10.1137","volume":"47","author":[{"given":"Scott","family":"Aaronson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andris","family":"Ambainis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,6,26]]},"reference":[{"key":"atypb1","first-page":"141","author":"Aaronson S.","year":"2010","journal-title":"Proc. ACM STOC"},{"key":"atypb2","first-page":"307","author":"Aaronson S.","year":"2015","journal-title":"Proc. ACM STOC"},{"key":"atypb3","unstructured":"S. Aaronson, A. Ambainis, J. Iraids, M. Kokainis, and J. Smotrovs,\n                      Polynomials, quantum query complexity, and Grothendieck's inequality\n                      , in Proc. 31st Conference on Computational Complexity (CCC '16), 2016, 25."},{"key":"atypb4","unstructured":"S. Aaronson and L. Chen,\n                      Complexity-theoretic foundations of quantum supremacy experiments\n                      , in Proc. 32nd Conference on Computational Complexity (CCC '17), 2017, 22."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.70.052328"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcss.2005.06.006"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1145\/502090.502097"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-002-0978-1"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539796300933"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539704442416"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00144-X"},{"key":"atypb12","unstructured":"S. Chakraborty, E. Fischer, A. Matsliah, and R. de Wolf,\n                      New results on quantum property testing\n                      , in Proc. Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 2010, pp. 145-156."},{"key":"atypb13","first-page":"59","author":"Childs A. M.","year":"2003","journal-title":"Proc. ACM STOC"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2004.04.001"},{"key":"atypb15","first-page":"526","author":"Cleve R.","year":"2000","journal-title":"Proc. IEEE FOCS"},{"key":"atypb16","first-page":"437","author":"Dinur I.","year":"2006","journal-title":"Proc. ACM STOC"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-012-0043-7"},{"key":"atypb18","unstructured":"S. Fenner and Y. Zhang,\n                      A Note on the Classical Lower Bound for a Quantum Walk Algorithm\n                      . preprint,https:\/\/arxiv.org\/abs\/quant-ph\/0312230, 2003."},{"key":"atypb19","first-page":"25","author":"Goldreich O.","year":"1989","journal-title":"Proc. ACM STOC"},{"key":"atypb20","first-page":"1","volume":"7","author":"Montanaro A.","year":"2016","journal-title":"Theory Comput. Grad. Surveys"},{"key":"atypb21","unstructured":"M. Nielsen and I. Chuang,\n                      Quantum Computation and Quantum Information\n                      , Cambridge University Press, Cambridge, UK, 2000."},{"key":"atypb22","unstructured":"R. O'Donnell and Y. Zhao,\n                      Polynomial bounds for decoupling, with applications\n                      , in Proc. 31st Conference on Computational Complexity (CCC '16), 2016, 24."},{"key":"atypb23","first-page":"358","author":"Raz R.","year":"1999","journal-title":"Proc. ACM STOC"},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"B. W. Reichardt,\n                      Reflections for quantum query algorithms\n                      , in Proc. 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2011, pp. 560-569,https:\/\/doi.org\/10.1137\/1.9781611973082.44.","DOI":"10.1137\/1.9781611973082.44"},{"key":"atypb25","unstructured":"O. Shamir,\n                      A Variant of Azuma's Inequality for Martingales with Subgaussian Tails,\n                      preprint,https:\/\/arxiv.org\/abs\/1110.2392, 2011."},{"key":"atypb26","first-page":"5","author":"Shepherd D. J.","year":"2006","journal-title":"Quantum Inf. Process."},{"key":"atypb27","first-page":"84","volume":"3","author":"Shi Y.","year":"2002","journal-title":"Quantum Inf. Comput."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539795293172"},{"key":"atypb29","first-page":"116","author":"Simon D.","year":"1994","journal-title":"Proc. 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