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DOI: 10.4230/LIPIcs.STACS.2013.434
Explicit relation between all lower bound techniques for quantum query complexity

Authors: Loïck Magnin and Jérémie Roland

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Abstract
The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.
Cite as

Loïck Magnin and Jérémie Roland. Explicit relation between all lower bound techniques for quantum query complexity. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 434-445, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{magnin_et_al:LIPIcs.STACS.2013.434,
  author =	{Magnin, Lo\"{i}ck and Roland, J\'{e}r\'{e}mie},
  title =	{{Explicit relation between all lower bound techniques for quantum query complexity}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{434--445},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.434},
  URN =		{urn:nbn:de:0030-drops-39548},
  doi =		{10.4230/LIPIcs.STACS.2013.434},
  annote =	{Keywords: Quantum computation, lower bound, adversary method, polynomial method}
}
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