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Barnum, Howard

Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries

06391.BarnumHoward.Paper.876.pdf (0.2 MB)


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.

BibTeX - Entry

  author =	{Howard Barnum},
  title =	{Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  year =	{2007},
  editor =	{Stephan Dahlke and Klaus Ritter and Ian H. Sloan and Joseph F. Traub},
  number =	{06391},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{},
  annote =	{Keywords: Quantum query complexity semidefinite programming}

Keywords: Quantum query complexity semidefinite programming
Seminar: 06391 - Algorithms and Complexity for Continuous Problems
Issue Date: 2007
Date of publication: 31.01.2007

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