### 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) |
---|---|

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 |

### Fingerprint

### 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

*Quantum Information and Computation*,

*14*(5-6), 517-540.

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

Research output: Contribution to journal › Article

*Quantum Information and Computation*, vol. 14, no. 5-6, pp. 517-540.

}

TY - JOUR

T1 - Hardness of approximation for quantum problems

AU - Gharibian, Sevag

AU - Kempe, Julia

PY - 2014/4/1

Y1 - 2014/4/1

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. 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.

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. 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.

KW - Hardness of approximation

KW - Polynomial time hierarchy

KW - Quantum complexity

KW - Succinct set cover

UR - http://www.scopus.com/inward/record.url?scp=84897756088&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84897756088&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84897756088

VL - 14

SP - 517

EP - 540

JO - Quantum Information and Computation

JF - Quantum Information and Computation

SN - 1533-7146

IS - 5-6

ER -