## Abstract

The k-LOCAL HAMILTONIAN problem is & natural complete problem for the complexity class QMA, the quantum analogue of NP. It is similar in spirit to MAX-k-SAT, which is NP-complete for k ≥ 2. It was known that the problem is QMA-complete for any k ≥ 3. On the other hand, 1-LOCAL HAMILTONIAN is in P and hence not believed to be QMA-complete. The complexity of the 2-LOCAL HAMILTONIAN problem has long been outstanding. Here we settle the question and show that it is QMA-complete. We provide two independent proofs; our first proof uses only elementary linear algebra. Our second proof uses a powerful technique for analyzing the sum of two Hamiltonians; this technique is based on perturbation theory and we believe that it might prove useful elsewhere. Using our techniques we also show that adiabatic computation with 2-local interactions on qubits is equivalent to standard quantum computation.

Original language | English (US) |
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Pages (from-to) | 1070-1097 |

Number of pages | 28 |

Journal | SIAM Journal on Computing |

Volume | 35 |

Issue number | 5 |

DOIs | |

State | Published - 2006 |

## Keywords

- Adiabatic computation
- Complete problems
- Local Hamiltonian problem
- Quantum computation

## ASJC Scopus subject areas

- General Computer Science
- General Mathematics