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

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 πΌππ’_=π― 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 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 Bouland et al (2021). We also use recent results in polynomial interpolation to show πΌππ’_=π― hardness under π‘π―π― reductions rather than π‘π―π―^ππ― 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 πΌππ’_=π― 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, which include Sherrington-Kirkpatrick and ErdΓΆs-Renyi graphs. For IQP circuits, a consequence of our results 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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