  Motivated by the recent experimental demonstrations of quantum supremacy,
proving the hardness of the output of random quantum circuits is an imperative
near term goal. We prove under the complexity theoretical assumption of the
non-collapse of the polynomial hierarchy that approximating the output
probabilities of random quantum circuits to within $\exp(-\Omega(m\log m))$
additive error is hard for any classical computer, where $m$ is the number of
gates in the quantum computation. More precisely, we show that the above
problem is $\#\mathsf{P}$-hard under $\mathsf{BPP}^{\mathsf{NP}}$ reduction. In
the recent experiments, the quantum circuit has $n$-qubits and the architecture
is a two-dimensional grid of size $\sqrt{n}\times\sqrt{n}$. Indeed for constant
depth circuits approximating the output probabilities to within
$2^{-\Omega(n\log{n})}$ is hard. For circuits of depth $\log{n}$ or $\sqrt{n}$
for which the anti-concentration property holds, approximating the output
probabilities to within $2^{-\Omega(n\log^2{n})}$ and $2^{-\Omega(n^{3/2}\log
n)}$ is hard respectively. We then show that the hardness results extend to any
open neighborhood of an arbitrary (fixed) circuit including the trivial circuit
with identity gates. We made an effort to find the best proofs and proved these
results from first principles, which do not use the standard techniques such as
the Berlekamp--Welch algorithm, the usual Paturi's lemma, and Rakhmanov's
result.
