Complexity classification of two-qubit commuting hamiltonians

Adam Bouland, Laura Mancinska, Xue Zhang

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

8 Citations (Scopus)


We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.

Original languageEnglish
Title of host publication31st Conference on Computational Complexity, CCC 2016
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770088
Publication statusPublished - 1 May 2016
Event31st Conference on Computational Complexity, CCC 2016 - Tokyo, Japan
Duration: 29 May 20161 Jun 2016


Conference31st Conference on Computational Complexity, CCC 2016


  • Commuting hamiltonians
  • Gate classification theorems
  • IQP
  • Quantum computing
  • Sampling problems


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