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The applications of random quantum circuits range from quantum computing andquantum many-body systems to the physics of black holes. Many of theseapplications are related to the generation of quantum pseudorandomness: Randomquantum 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 thatrandom quantum circuits on qubits in a brickwork architecture of depth $O(nt^{10.5})$ are approximate unitary $t$-designs. In this work, we revisit thisargument, which lower bounds the spectral gap of moment operators for localrandom quantum circuits by $\Omega(n^{-1}t^{-9.5})$. We improve this lowerbound 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 quantumcircuits generate approximate unitary $t$-designs in depth $O(nt^{5+o(1)})$.Our techniques involve Gao's quantum union bound and the unreasonableeffectiveness of the Clifford group. As an auxiliary result, we prove fastconvergence to the Haar measure for random Clifford unitaries interleaved withHaar random single qubit unitaries.

Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$