Additive-error fine-grained quantum supremacy

Morimae, Tomoyuki; Tamaki, Suguru

doi:10.22331/q-2020-09-24-329

Cited by 4 publications

(1 citation statement)

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“…To go beyond asymptotic hardness results, [31], [32] use conjectures in so-called fine-grained complexity to obtain results about the number of qubits needed to show quantum supremacy. Finally, while we do not study it in our paper, an interesting sampling problem called Fourier sampling defined for a particular class of circuits, has been studied in [33].…”

Section: Introductionmentioning

confidence: 99%

Average-case hardness of estimating probabilities of random quantum circuits with a linear scaling in the error exponent

Krovi¹

2022

Preprint

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We consider the hardness of computing additive approximations to output probabilities of random quantum circuits. We consider three random circuit families, namely, Haar random, p = 1 QAOA, and random IQP circuits. Our results are as follows.For Haar random circuits with m gates, we improve on prior results by showing coC = P hardness of average-case additive approximations to an imprecision of 2 −O(m) . Efficient classical simulation of such problems would imply the collapse of the polynomial hierarchy. For constant depth circuits i.e., when m = O(n), this linear scaling in the exponent is to within a constant of the scaling required to show hardness of sampling. Prior to our work, such a result was shown only for Boson Sampling in [1]. We also use recent results in polynomial interpolation to show coC = P hardness under BPP reductions rather than BPP NP reductions. This improves the results of prior work for Haar random circuits both in terms of the error scaling and the power of reductions.Next, we consider random p = 1 QAOA and IQP circuits and show that in the average-case, it is coC = P hard to approximate the output probability to within an additive error of 2 −O(n) . For p = 1 QAOA circuits, this work constitutes the first average-case hardness result for the problem of approximating output probabilities for random QAOA circuits. The random QAOA circuits we consider include those coming from Sherrington-Kirkpatrick models and Erdös-Renyi graphs. For IQP circuits, we prove our results without additional conjectures on the hardness of Ising partition functions. Indeed, a consequence of our results for IQP circuits is that approximating the Ising partition function with imaginary couplings to an additive error of 2 −O(n) is hard even in the average-case, which extends prior work on worst-case hardness of multiplicative approximation to Ising partition functions.

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Section: Introductionmentioning

confidence: 99%