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

by   Hari Krovi, et al.

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 π–Όπ—ˆπ–’_=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 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 π–Όπ—ˆπ–’_=𝖯 hardness under 𝖑𝖯𝖯 reductions rather than 𝖑𝖯𝖯^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 π–Όπ—ˆπ–’_=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, 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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