## Abstract

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 language | English |
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Title of host publication | 31st Conference on Computational Complexity, CCC 2016 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 28:1-28:33 |

Volume | 50 |

ISBN (Electronic) | 9783959770088 |

DOIs | |

Publication status | Published - 1 May 2016 |

Event | 31st Conference on Computational Complexity, CCC 2016 - Tokyo, Japan Duration: 29 May 2016 → 1 Jun 2016 |

### Conference

Conference | 31st Conference on Computational Complexity, CCC 2016 |
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Country/Territory | Japan |

City | Tokyo |

Period | 29/05/16 → 1/06/16 |

## Keywords

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