{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article1591","name":"Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries","abstract":"Generalizing earlier work characterizing the quantum query\r\ncomplexity of computing a function of an unknown classical ``black box''\r\nfunction\r\ndrawn from some set of such black box functions, \r\nwe investigate a more general quantum query model in which \r\nthe goal is to compute\r\nfunctions of $N \times N$ ``black box'' unitary matrices drawn from \r\na set of such matrices, a problem with\r\napplications to determining properties of quantum physical systems. \r\nWe characterize the existence of an algorithm for such a query problem, \r\nwith given query and error, as equivalent to the feasibility of a certain set of semidefinite\r\nprogramming constraints, or equivalently the infeasibility of a dual of these \r\nconstraints, which we construct. Relaxing the primal constraints to correspond \r\nto mere pairwise near-orthogonality of the final states of a quantum computer, conditional\r\non the various black-box inputs, rather than bounded-error distinguishability,\r\nwe obtain a relaxed primal program the feasibility of\r\nwhose dual still implies the nonexistence of a quantum algorithm. We use this to obtain\r\na generalization, to our not-necessarily-commutative setting,\r\nof the ``spectral adversary method'' for quantum query lower bounds.","keywords":"Quantum query complexity semidefinite programming","author":{"@type":"Person","name":"Barnum, Howard","givenName":"Howard","familyName":"Barnum"},"position":3,"pageStart":1,"pageEnd":25,"dateCreated":"2007-01-31","datePublished":"2007-01-31","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Barnum, Howard","givenName":"Howard","familyName":"Barnum"},"copyrightYear":"2007","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06391.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume625","volumeNumber":6391,"name":"Dagstuhl Seminar Proceedings, Volume 6391","dateCreated":"2007-01-31","datePublished":"2007-01-31","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article1591","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume625"}}}