  Sampling from the output distribution of chaotic quantum evolutions, and of
pseudo-random universal quantum circuits in particular, has been proposed as a
prominent milestone for near-term quantum supremacy. The same paper notes that
chaotic distributions are very sensitive to noise, and under quite general
noise models converge to the uniform distribution over bit-strings
exponentially in the number of gates. On the one hand, for increasing number of
gates, it suffices to choose bit-strings at random to approximate the noisy
distribution with fixed statistical distance. On the other hand, cross-entropy
benchmarking can be used to gauge the fidelity of an experiment, and the
distance to the uniform distribution. We estimate that state-of-the-art
classical supercomputers would fail to simulate high-fidelity chaotic quantum
circuits with approximately fifty qubits and depth forty. A recent interesting
paper proposed a different approximation algorithm to a noisy distribution,
extending previous results on the Fourier analysis of commuting quantum
circuits. Using the statistical properties of the Porter-Thomas distribution,
we show that this new approximation algorithm does not improve random guessing,
in polynomial time. Therefore, it confirms previous results and does not
represent an additional challenge to the suggested failure stated above.
