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The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures.
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PP is as hard as the polynomial-time hierarchy
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D. Shepherd and M. J. Bremner · 2009
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M. Bremner, R. Jozsa, and D. Shepherd · 2011
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T. Morimae, K. Fujii, and J. Fitzsimons · 2014
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