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 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 languageEnglish (US)
Pages (from-to)372-383
Number of pages12
JournalLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume3328
StatePublished - 2004

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

ASJC Scopus subject areas

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

Cite this

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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.",
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