## 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 language | English (US) |
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Pages (from-to) | 517-540 |

Number of pages | 24 |

Journal | Quantum Information and Computation |

Volume | 14 |

Issue number | 5-6 |

State | Published - Apr 1 2014 |

## Keywords

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

## ASJC Scopus subject areas

- Theoretical Computer Science
- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics
- General Physics and Astronomy
- Computational Theory and Mathematics