The complexity of the local hamiltonian problem

Julia Kempe, Alexei Kitaev, Oded Regev

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1070-1097
Number of pages28
JournalSIAM Journal on Computing
Volume35
Issue number5
DOIs
StatePublished - 2006

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Hamiltonians
Quantum computers
Linear algebra
Local Interaction
Quantum Computation
Complexity Classes
Qubit
Perturbation Theory
NP-complete problem
Analogue

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

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

In: SIAM Journal on Computing, Vol. 35, No. 5, 2006, p. 1070-1097.

Research output: Contribution to journalArticle

Kempe, Julia ; Kitaev, Alexei ; Regev, Oded. / The complexity of the local hamiltonian problem. In: SIAM Journal on Computing. 2006 ; Vol. 35, No. 5. pp. 1070-1097.
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