License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.MFCS.2018.58
URN: urn:nbn:de:0030-drops-96409
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9640/
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Gharibian, Sevag ; Santha, Miklos ; Sikora, Jamie ; Sundaram, Aarthi ; Yirka, Justin

Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)

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LIPIcs-MFCS-2018-58.pdf (0.6 MB)

Abstract

The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH does not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH, uses classical proofs, and the second, QPH, uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, Q Sigma_3, into NEXP using the Ellipsoid Method for efficiently solving semidefinite programs. These results yield two implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if QCPH=QPH (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs "equivalent"), then QMA(2) is in the Counting Hierarchy (specifically, in P^{PP^{PP}}). Second, unless QMA(2)= Q Sigma_3 (i.e., alternating quantifiers do not help in the presence of "unentanglement"), QMA(2) is strictly contained in NEXP.

BibTeX - Entry

@InProceedings{gharibian_et_al:LIPIcs:2018:9640,
  author =	{Sevag Gharibian and Miklos Santha and Jamie Sikora and Aarthi Sundaram and Justin Yirka},
  title =	{{Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)}},
  booktitle =	{43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)},
  pages =	{58:1--58:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Igor Potapov and Paul Spirakis and James Worrell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9640},
  URN =		{urn:nbn:de:0030-drops-96409},
  doi =		{10.4230/LIPIcs.MFCS.2018.58},
  annote =	{Keywords: Complexity Theory, Quantum Computing, Polynomial Hierarchy, Semidefinite Programming, QMA(2), Quantum Complexity}
}

Keywords: Complexity Theory, Quantum Computing, Polynomial Hierarchy, Semidefinite Programming, QMA(2), Quantum Complexity
Collection: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
Issue Date: 2018
Date of publication: 27.08.2018

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