### 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 language | English (US) |
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Title of host publication | Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings |

Pages | 387-398 |

Number of pages | 12 |

Edition | PART 1 |

DOIs | |

State | Published - Dec 1 2012 |

Event | 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012 - Warwick, United Kingdom Duration: Jul 9 2012 → Jul 13 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 1 |

Volume | 7391 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012 |
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Country | United Kingdom |

City | Warwick |

Period | 7/9/12 → 7/13/12 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - Hardness of approximation for quantum problems

AU - Gharibian, Sevag

AU - Kempe, Julia

PY - 2012/12/1

Y1 - 2012/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84883794770&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883794770&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-31594-7_33

DO - 10.1007/978-3-642-31594-7_33

M3 - Conference contribution

SN - 9783642315930

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 387

EP - 398

BT - Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings

ER -