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Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries Barnum, Howard Quantum query complexity semidefinite programming Generalizing earlier work characterizing the quantum query complexity of computing a function of an unknown classical ``black box'' function drawn from some set of such black box functions, we investigate a more general quantum query model in which the goal is to compute functions of $N imes N$ ``black box'' unitary matrices drawn from a set of such matrices, a problem with applications to determining properties of quantum physical systems. We characterize the existence of an algorithm for such a query problem, with given query and error, as equivalent to the feasibility of a certain set of semidefinite programming constraints, or equivalently the infeasibility of a dual of these constraints, which we construct. Relaxing the primal constraints to correspond to mere pairwise near-orthogonality of the final states of a quantum computer, conditional on the various black-box inputs, rather than bounded-error distinguishability, we obtain a relaxed primal program the feasibility of whose dual still implies the nonexistence of a quantum algorithm. We use this to obtain a generalization, to our not-necessarily-commutative setting, of the ``spectral adversary method'' for quantum query lower bounds. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Howard Barnum 2007 Is Part Of Dagstuhl Seminar Proceedings, Volume 6391, Algorithms and Complexity for Continuous Problems (2007) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.06391.3 urn:nbn:de:0030-drops-8769 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06391.3 eng https://creativecommons.org/licenses/by/4.0/legalcode