## Abstract

In this paper, we study variants of the canonical Local Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local Hamiltonian problem. The input for the Separable Local Hamiltonian problem is the same as the Local Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable Sparse Hamiltonian problem is defined similarly, but the Hamiltonian is not necessarily local, but rather sparse. We show that the Separable Sparse Hamiltonian problem is QMA(2)-Complete, while Separable Local Hamiltonian is in QMA. This should be compared to the Local Hamiltonian problem, and the Sparse Hamiltonian problem which are both QMA-Complete. To the best of our knowledge, Separable Sparse Hamiltonian is the first non-trivial problem shown to be QMA(2)-Complete.

Original language | English |
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Title of host publication | Proceedings - 2012 IEEE 27th Conference on Computational Complexity, CCC 2012 |

Pages | 32-41 |

Number of pages | 10 |

DOIs | |

State | Published - 26 Sep 2012 |

Externally published | Yes |

Event | IEEE Computer Society Technical Committee on Mathematical Foundations of Computing - Porto, Portugal Duration: 26 Jun 2012 → 29 Jun 2012 |

### Conference

Conference | IEEE Computer Society Technical Committee on Mathematical Foundations of Computing |
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Country/Territory | Portugal |

City | Porto |

Period | 26/06/12 → 29/06/12 |

## ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Computational Mathematics