Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy

@article{Bremner2010ClassicalSO,
  title={Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy},
  author={Michael J. Bremner and Richard Jozsa and Dan J. Shepherd},
  journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
  year={2010},
  volume={467},
  pages={459 - 472}
}
  • M. BremnerR. JozsaD. Shepherd
  • Published 9 May 2010
  • Computer Science, Mathematics
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
We consider quantum computations comprising only commuting gates, known as IQP computations, and provide compelling evidence that the task of sampling their output probability distributions is unlikely to be achievable by any efficient classical means. More specifically, we introduce the class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection, and prove first that post-IQP equals the classical class PP. Using this result we show that if the… 

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