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

Pages (from-to) | 1070-1097 |

Number of pages | 28 |

Journal | SIAM Journal on Computing |

Volume | 35 |

Issue number | 5 |

DOIs | |

State | Published - 2006 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Computing*,

*35*(5), 1070-1097. https://doi.org/10.1137/S0097539704445226

**The complexity of the local hamiltonian problem.** / Kempe, Julia; Kitaev, Alexei; Regev, Oded.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 35, no. 5, pp. 1070-1097. https://doi.org/10.1137/S0097539704445226

}

TY - JOUR

T1 - The complexity of the local hamiltonian problem

AU - Kempe, Julia

AU - Kitaev, Alexei

AU - Regev, Oded

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Adiabatic computation

KW - Complete problems

KW - Local Hamiltonian problem

KW - Quantum computation

UR - http://www.scopus.com/inward/record.url?scp=33750165209&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750165209&partnerID=8YFLogxK

U2 - 10.1137/S0097539704445226

DO - 10.1137/S0097539704445226

M3 - Article

AN - SCOPUS:33750165209

VL - 35

SP - 1070

EP - 1097

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 5

ER -