### Abstract

The k-LOCAL HAMILTONIAN problem is a natural complete problem for the complexity class QMA, the quantum analog 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 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. The second proof uses elementary linear algebra only. Using our techniques we also show that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation.

Original language | English (US) |
---|---|

Pages (from-to) | 372-383 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 3328 |

State | Published - 2004 |

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### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*3328*, 372-383.

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

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 3328, pp. 372-383.

}

TY - JOUR

T1 - The complexity of the local Hamiltonian problem

AU - Kempe, Julia

AU - Kitaev, Alexei

AU - Regev, Oded

PY - 2004

Y1 - 2004

N2 - The k-LOCAL HAMILTONIAN problem is a natural complete problem for the complexity class QMA, the quantum analog 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 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. The second proof uses elementary linear algebra only. Using our techniques we also show that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation.

AB - The k-LOCAL HAMILTONIAN problem is a natural complete problem for the complexity class QMA, the quantum analog 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 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. The second proof uses elementary linear algebra only. Using our techniques we also show that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation.

UR - http://www.scopus.com/inward/record.url?scp=34147163913&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34147163913&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:34147163913

VL - 3328

SP - 372

EP - 383

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -