Hardness of Approximation for Quantum Problems

  • Sevag Gharibian
  • Julia Kempe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)


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.


Quantum Circuit Quantum Problem Quantum Setting Polynomial Hierarchy Quantum Generalization 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sevag Gharibian
    • 1
  • Julia Kempe
    • 2
    • 3
  1. 1.David R. Cheriton School of Computer Science and Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  2. 2.CNRS & LIAFA, University Paris Diderot - Paris 7ParisFrance
  3. 3.Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael

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