Understanding the power of quantum supremacy experiments is one of the most pressing topics in the NISQ era. In this work we make progress toward bridging the remaining gaps between theory and experiment, incorporating the effects of experimental noise into the theoretical hardness arguments. First, we improve the robustness of prior hardness results in this area. For random circuit sampling, we show that computing the output probability of an $m$-gate random quantum circuit to within additive imprecision $2^{-O(m\log m)}$ is $\#\mathsf{P}$-hard, an exponential improvement over the prior hardness results of Bouland et al. and Movassagh which were resistant to imprecision $2^{-O(m^3)}$. This nearly reaches the threshold ($2^{-O(m)}$) sufficient to establish the hardness of sampling for constant-depth random circuits. For BosonSampling, we show $\#\mathsf{P}$-hardness of computing output probabilities of $n$-photon $m=\text{poly}(n)$ mode BosonSampling experiments to additive error $2^{-O(n\log n)}$. This is within a constant factor in the exponent of the robustness required for hardness of sampling. To prove these results we introduce a new robust Berlekamp-Welch argument over the Reals, and moreover substantially simplify prior robustness arguments. Second, we show that in the random circuit sampling case, these results are still true in the presence of a constant rate of noise, so long as the noise rate is below the error detection threshold. That is, even though random circuits with a constant noise rate converge rapidly to the maximally mixed state, the small deviations in their output probabilities away from uniform remain difficult to compute. Interestingly, we then show that our results are in tension with one another, and the latter result implies the former is essentially optimal with respect to additive imprecision, even with generalizations of our techniques.

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