Hardness of approximation for quantum problems

  • Authors:

  • Sevag Gharibian;Julia Kempe

  • Affiliations:

  • David R. Cheriton School of Computer Science and Institute for Quantum Computing, University of Waterloo, Waterloo, Canada;CNRS & LIAFA, University Paris Diderot - Paris 7, Paris, France, Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel

  • Venue:
  • ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
  • Year:
  • 2012

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Abstract

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Our work thus yields the first known hardness of approximation results for a quantum complexity class. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy (Umans 1999). We close by showing that a variant of the local Hamiltonian problem with hybrid classical-quantum ground states is complete for the second level of our quantum hierarchy.