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

• URL: http://arxiv.org/abs/2206.05642v1
• Date: Sun, 12 Jun 2022 02:35:51 GMT
• Title: Average-case hardness of estimating probabilities of random quantum circuits with a linear scaling in the error exponent
• Authors: Hari Krovi
• Abstract summary: We consider the hardness of computing additive approximations to output probabilities of random quantum circuits. For Haar random circuits with $m$ gates, we show $mathsfcoC_=P$ hardness of average-case additive approximations to an imprecision of $2-O(m)$. For random $p=1$ QAOA and IQP circuits, we show that in the average-case, it is $mathsfcoC_=P$ hard to approximate the output probability to within an additive error of $2-O(n)$
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\mathsf{coC_=P}$ 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 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 $\mathsf{coC_=P}$ hardness under $\mathsf{BPP}$ reductions rather than $\mathsf{BPP}^{\mathsf{NP}}$ 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 $\mathsf{coC_=P}$ 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, which include Sherrington-Kirkpatrick and Erd\"{o}s-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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