  The applications of random quantum circuits range from quantum computing and
quantum many-body systems to the physics of black holes. Many of these
applications are related to the generation of quantum pseudorandomness: Random
quantum circuits are known to approximate unitary $t$-designs. Unitary
$t$-designs are probability distributions that mimic Haar randomness up to
$t$th moments. In a seminal paper, Brand\~{a}o, Harrow and Horodecki prove that
random quantum circuits on qubits in a brickwork architecture of depth $O(n
t^{10.5})$ are approximate unitary $t$-designs. In this work, we revisit this
argument, which lower bounds the spectral gap of moment operators for local
random quantum circuits by $\Omega(n^{-1}t^{-9.5})$. We improve this lower
bound to $\Omega(n^{-1}t^{-4-o(1)})$, where the $o(1)$ term goes to $0$ as
$t\to\infty$. A direct consequence of this scaling is that random quantum
circuits generate approximate unitary $t$-designs in depth $O(nt^{5+o(1)})$.
Our techniques involve Gao's quantum union bound and the unreasonable
effectiveness of the Clifford group. As an auxiliary result, we prove fast
convergence to the Haar measure for random Clifford unitaries interleaved with
Haar random single qubit unitaries.
