Hardness of approximation for quantum problems

Sevag Gharibian, Julia Kempe

Research output: Contribution to journalArticle

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. Using the same 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, FOCS 1999]. The problems for which we prove hardness of approximation for include, among others, a quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian problem with hybrid classical-quantum ground states.

Original languageEnglish (US)
Pages (from-to)517-540
Number of pages24
JournalQuantum Information and Computation
Volume14
Issue number5-6
StatePublished - Apr 1 2014

Fingerprint

Hardness of Approximation
hierarchies
Polynomial Hierarchy
hardness
Hardness
Polynomials
polynomials
approximation
Hamiltonians
Ground state
Set Cover
Complexity Theory
Ground State
ground state

Keywords

  • Hardness of approximation
  • Polynomial time hierarchy
  • Quantum complexity
  • Succinct set cover

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Nuclear and High Energy Physics

Cite this

Gharibian, S., & Kempe, J. (2014). Hardness of approximation for quantum problems. Quantum Information and Computation, 14(5-6), 517-540.

Hardness of approximation for quantum problems. / Gharibian, Sevag; Kempe, Julia.

In: Quantum Information and Computation, Vol. 14, No. 5-6, 01.04.2014, p. 517-540.

Research output: Contribution to journalArticle

Gharibian, S & Kempe, J 2014, 'Hardness of approximation for quantum problems', Quantum Information and Computation, vol. 14, no. 5-6, pp. 517-540.
Gharibian, Sevag ; Kempe, Julia. / Hardness of approximation for quantum problems. In: Quantum Information and Computation. 2014 ; Vol. 14, No. 5-6. pp. 517-540.
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