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 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.

Discord in relation to resource states for measurement-based quantum computation

“…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].…”

“…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.…”

“…The foundational proposal [1] for MBQC introduced the class of cluster states for the initial states, but since then many other types of entangled states have been found that support universal quantum computation [2][3][4]. Universality is obviously a sufficient condition for a state to be a resource state for MBQC, but seems too strong as a necessary condition given the recent discovery of classes of MBQCs such as IQP [5]. This class includes only non-adaptive MBQCs, and so is unlikely to be universal, but nevertheless appears to give an advantage over classical computing.…”

Ancilla-driven instantaneous quantum polynomial time circuit for quantum supremacy

“…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.…”

“…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.…”