Journal
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume 467, Issue 2126, Pages 459-472Publisher
ROYAL SOC
DOI: 10.1098/rspa.2010.0301
Keywords
quantum computational complexity; quantum simulation; polynomial hierarchy
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Funding
- EU [233747]
- National Science Foundation [PHY05-51164]
- UK EPSRC QIPIRC
- EC network QICS
- CESG
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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 output distributions of uniform IQP circuit families could be classically efficiently sampled, either exactly in total variation distance or even approximately up to 41 per cent multiplicative error in the probabilities, then the infinite tower of classical complexity classes known as the polynomial hierarchy would collapse to its third level. We mention some further results on the classical simulation properties of IQP circuit families, in particular showing that if the output distribution results from measurements on only O(log n) lines then it may, in fact, be classically efficiently sampled.
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