Random reversible and quantum circuits form random walks on the alternating group $\mathrm{Alt}(2^n)$ and unitary group $\mathrm{SU}(2^n)$, respectively. Known bounds on the spectral gap for the $t$-th moment of these random walks have inverse-polynomial dependence in both $n$ and $t$. We prove that the gap for random reversible circuits is $\Omega(n^{-3})$ for all $t\geq 1$, and the gap for random quantum circuits is $\Omega(n^{-3})$ for $t \leq \Theta(2^{n/2})$. These gaps are independent of $t$ in the respective regimes. We can further improve both gaps to $n^{-1}/\mathrm{polylog}(n, t)$ for $t\leq 2^{\Theta(n)}$, which is tight up to polylog factors. Our spectral gap results have a number of consequences: 1) Random reversible circuits with $\mathcal{O}(n^4 t)$ gates form multiplicative-error $t$-wise independent (even) permutations for all $t\geq 1$; for $t \leq \Theta(2^{n/6.1})$, we show that $\tilde{\mathcal{O}}(n^2 t)$ gates suffice. 2) Random quantum circuits with $\mathcal{O}(n^4 t)$ gates form multiplicative-error unitary $t$-designs for $t \leq \Theta(2^{n/2})$; for $t\leq \Theta(2^{2n/5})$, we show that $\tilde{\mathcal{O}}(n^2t)$ gates suffice. 3) The robust quantum circuit complexity of random circuits grows linearly for an exponentially long time, proving the robust Brown--Susskind conjecture [BS18,BCHJ+21]. Our spectral gap bounds are proven by reducing random quantum circuits to a more structured walk: a modification of the ``$\mathrm{PFC}$ ensemble'' from [MPSY24] together with an expander on the alternating group due to Kassabov [Kas07a], for which we give an efficient implementation using reversible circuits. In our reduction, we approximate the structured walk with local random circuits without losing the gap, which uses tools from the study of frustration-free Hamiltonians.

翻译：随机可逆电路与随机量子电路分别在交错群 $\mathrm{Alt}(2^n)$ 和酉群 $\mathrm{SU}(2^n)$ 上形成随机游走。已知这些随机游走 $t$ 阶矩的谱隙下界对 $n$ 和 $t$ 均具有逆多项式依赖关系。我们证明：对于所有 $t\geq 1$，随机可逆电路的谱隙为 $\Omega(n^{-3})$；对于 $t \leq \Theta(2^{n/2})$，随机量子电路的谱隙为 $\Omega(n^{-3})$。这些谱隙在各自参数范围内均与 $t$ 无关。进一步地，当 $t\leq 2^{\Theta(n)}$ 时，我们可以将两类电路的谱隙改进至 $n^{-1}/\mathrm{polylog}(n, t)$，该结果在多项式对数因子内是紧的。我们的谱隙结论具有以下重要推论：1）具有 $\mathcal{O}(n^4 t)$ 门数量的随机可逆电路可生成乘性误差的 $t$ 阶独立（偶）置换（对所有 $t\geq 1$）；当 $t \leq \Theta(2^{n/6.1})$ 时，我们证明 $\tilde{\mathcal{O}}(n^2 t)$ 门数量即足够。2）具有 $\mathcal{O}(n^4 t)$ 门数量的随机量子电路可在 $t \leq \Theta(2^{n/2})$ 时生成乘性误差的酉 $t$-设计；当 $t\leq \Theta(2^{2n/5})$ 时，我们证明 $\tilde{\mathcal{O}}(n^2t)$ 门数量即足够。3）随机电路的鲁棒量子电路复杂度在指数级时长内呈线性增长，从而证明了鲁棒性 Brown--Susskind 猜想 [BS18,BCHJ+21]。我们的谱隙下界证明通过将随机量子电路规约至一种更具结构性的游走实现：该游走结合了 [MPSY24] 中“$\mathrm{PFC}$ 系综”的改进形式，以及 Kassabov [Kas07a] 提出的交错群上的扩展图（我们通过可逆电路给出了其高效实现）。在规约过程中，我们利用局部随机电路逼近该结构化游走且不损失谱隙，这一过程运用了无阻挫哈密顿量研究中的工具。