{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T19:47:52Z","timestamp":1774122472458,"version":"3.50.1"},"reference-count":42,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,4,8]],"date-time":"2021-04-08T00:00:00Z","timestamp":1617840000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"IdEx Universit\u00e9 de Paris","award":["ANR-18-IDEX-0001"],"award-info":[{"award-number":["ANR-18-IDEX-0001"]}]},{"DOI":"10.13039\/501100001665","name":"French National Research Agency","doi-asserted-by":"crossref","award":["QuBIC"],"award-info":[{"award-number":["QuBIC"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001665","name":"French National Research Agency","doi-asserted-by":"crossref","award":["QuDATA"],"award-info":[{"award-number":["QuDATA"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"We present a quantum interior-point method (IPM) for second-order cone programming (SOCP) that runs in time O<\/mml:mi>~<\/mml:mo><\/mml:mover><\/mml:mrow>(<\/mml:mo>n<\/mml:mi>r<\/mml:mi><\/mml:msqrt>\u03b6<\/mml:mi>\u03ba<\/mml:mi><\/mml:mrow>\u03b4<\/mml:mi>2<\/mml:mn><\/mml:msup><\/mml:mfrac>log<\/mml:mi>\u2061<\/mml:mo>(<\/mml:mo>1<\/mml:mn>\/<\/mml:mo><\/mml:mrow>\u03f5<\/mml:mi><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:math> where r<\/mml:mi><\/mml:math> is the rank and n<\/mml:mi><\/mml:math> the dimension of the SOCP, \u03b4<\/mml:mi><\/mml:math> bounds the distance of intermediate solutions from the cone boundary, \u03b6<\/mml:mi><\/mml:math> is a parameter upper bounded by n<\/mml:mi><\/mml:msqrt><\/mml:math>, and \u03ba<\/mml:mi><\/mml:math> is an upper bound on the condition number of matrices arising in the classical IPM for SOCP. The algorithm takes as its input a suitable quantum description of an arbitrary SOCP and outputs a classical description of a \u03b4<\/mml:mi><\/mml:math>-approximate \u03f5<\/mml:mi><\/mml:math>-optimal solution of the given problem.Furthermore, we perform numerical simulations to determine the values of the aforementioned parameters when solving the SOCP up to a fixed precision \u03f5<\/mml:mi><\/mml:math>. We present experimental evidence that in this case our quantum algorithm exhibits a polynomial speedup over the best classical algorithms for solving general SOCPs that run in time O<\/mml:mi>(<\/mml:mo>n<\/mml:mi>\u03c9<\/mml:mi>+<\/mml:mo>0.5<\/mml:mn><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:math> (here, \u03c9<\/mml:mi><\/mml:math> is the matrix multiplication exponent, with a value of roughly 2.37<\/mml:mn><\/mml:math> in theory, and up to 3<\/mml:mn><\/mml:math> in practice). For the case of random SVM (support vector machine) instances of size O<\/mml:mi>(<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:math>, the quantum algorithm scales as O<\/mml:mi>(<\/mml:mo>n<\/mml:mi>k<\/mml:mi><\/mml:msup>)<\/mml:mo><\/mml:math>, where the exponent k<\/mml:mi><\/mml:math> is estimated to be 2.59<\/mml:mn><\/mml:math> using a least-squares power law. On the same family random instances, the estimated scaling exponent for an external SOCP solver is 3.31<\/mml:mn><\/mml:math> while that for a state-of-the-art SVM solver is 3.11<\/mml:mn><\/mml:math>.<\/jats:p>","DOI":"10.22331\/q-2021-04-08-427","type":"journal-article","created":{"date-parts":[[2021,4,8]],"date-time":"2021-04-08T14:00:00Z","timestamp":1617890400000},"page":"427","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":23,"title":["Quantum algorithms for Second-Order Cone Programming and Support Vector Machines"],"prefix":"10.22331","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0659-3727","authenticated-orcid":false,"given":"Iordanis","family":"Kerenidis","sequence":"first","affiliation":[{"name":"QCWare, Palo Alto, California"},{"name":"Universit\u00e9 de Paris, CNRS, IRIF, F-75006, Paris, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6853-7125","authenticated-orcid":false,"given":"Anupam","family":"Prakash","sequence":"additional","affiliation":[{"name":"QCWare, Palo Alto, California"},{"name":"Universit\u00e9 de Paris, CNRS, IRIF, F-75006, Paris, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8518-8422","authenticated-orcid":false,"given":"D\u00e1niel","family":"Szil\u00e1gyi","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Paris, CNRS, IRIF, F-75006, Paris, France"}]}],"member":"9598","published-online":{"date-parts":[[2021,4,8]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"F. 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