2010
DOI: 10.1098/rspa.2010.0301
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Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy

Abstract: 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 distribu… Show more

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Cited by 349 publications
(590 citation statements)
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“…Before presenting our argument in more detail, it is worth mentioning that we have no disagreement with the other main claim of Hoban et al [9], namely their Lemma 1, in which they have applied similar proof techniques to those in [5] to show that efficient classical sampling of the output of IQP * circuits, a subclass of IQP circuits with a more standard uniformity condition, implies the collapse of the polynomial hierarchy to the third level, just as it does for IQP circuits [5]. Other sampling problems that can be achieved efficiently using quantum means and are unlikely to be efficiently achievable classically are known [15][16][17].…”
Section: Discussion Of Hoban Et Al's Claimsmentioning
confidence: 84%
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“…Before presenting our argument in more detail, it is worth mentioning that we have no disagreement with the other main claim of Hoban et al [9], namely their Lemma 1, in which they have applied similar proof techniques to those in [5] to show that efficient classical sampling of the output of IQP * circuits, a subclass of IQP circuits with a more standard uniformity condition, implies the collapse of the polynomial hierarchy to the third level, just as it does for IQP circuits [5]. Other sampling problems that can be achieved efficiently using quantum means and are unlikely to be efficiently achievable classically are known [15][16][17].…”
Section: Discussion Of Hoban Et Al's Claimsmentioning
confidence: 84%
“…Universality is certainly a sufficient condition, but seems too strong as a necessary condition for a state to be a resource for MBQC in light of results such as the following: (i) Anders and Browne [14] showed that single-qubit measurements on GHZ states can boost the extremely limited classical computational class ⊕L (parity-L) to P . (ii) Bremner et al [5] showed that IQP contains computations not in P (unless the polynomial hierarchy collapses to the third level) while unlikely to give universal quantum computation. Here IQP is the class of "instantaneous quantum computations," that is, those that can be carried out with nonadaptive measurements in the MBQC model.…”
Section: A Key Principlementioning
confidence: 99%
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“…It has been proven that IQP circuits, DQC1 circuits, and boson samplers are unlikely to be simulated classically up to a multiplicative error [3,10,18]. However, since a multiplicative error is unnatural, it is also desirable to show such unlikeliness in the case of an error in the l 1 norm.…”
Section: Introductionmentioning
confidence: 99%
“…Recently, in order to develop an understanding of the difference between them, the classical simulatability of restricted quantum computation has been extensively studied. For such quantum computation, commuting quantum computation including instantaneous quantum polynomial time (IQP) [2][3][4][5][6][7], deterministic quantum computation with 1 pure qubit (DQC1) [8][9][10], boson sampling [11][12][13][14], constant-depth quantum circuit [15,16], and permutational quantum computing [17] have been proposed.…”
Section: Introductionmentioning
confidence: 99%