Hardness of approximation for quantum problems

Sevag Gharibian, Julia Kempe

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish (US)
Title of host publicationAutomata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings
Pages387-398
Number of pages12
EditionPART 1
DOIs
StatePublished - Dec 1 2012
Event39th International Colloquium on Automata, Languages, and Programming, ICALP 2012 - Warwick, United Kingdom
Duration: Jul 9 2012Jul 13 2012

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume7391 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other39th International Colloquium on Automata, Languages, and Programming, ICALP 2012
CountryUnited Kingdom
CityWarwick
Period7/9/127/13/12

Fingerprint

Hardness of Approximation
Polynomial Hierarchy
Hardness
Polynomials
Hamiltonians
Ground state
Complexity Theory
Complexity Classes
Ground State

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Gharibian, S., & Kempe, J. (2012). Hardness of approximation for quantum problems. In Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings (PART 1 ed., pp. 387-398). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7391 LNCS, No. PART 1). https://doi.org/10.1007/978-3-642-31594-7_33

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

Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings. PART 1. ed. 2012. p. 387-398 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7391 LNCS, No. PART 1).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Gharibian, S & Kempe, J 2012, Hardness of approximation for quantum problems. in Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings. PART 1 edn, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 1, vol. 7391 LNCS, pp. 387-398, 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012, Warwick, United Kingdom, 7/9/12. https://doi.org/10.1007/978-3-642-31594-7_33
Gharibian S, Kempe J. Hardness of approximation for quantum problems. In Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings. PART 1 ed. 2012. p. 387-398. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); PART 1). https://doi.org/10.1007/978-3-642-31594-7_33
Gharibian, Sevag ; Kempe, Julia. / Hardness of approximation for quantum problems. Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings. PART 1. ed. 2012. pp. 387-398 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); PART 1).
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