  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.
