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：随机可逆电路和量子电路分别在交替群 $\mathrm{Alt}(2^n)$ 和酉群 $\mathrm{SU}(2^n)$ 上形成随机游走。关于这些随机游走的 $t$ 阶矩谱间隙的已知界在 $n$ 和 $t$ 上均呈逆多项式依赖。我们证明：对于所有 $t \geq 1$，随机可逆电路的谱间隙为 $\Omega(n^{-3})$；对于 $t \leq \Theta(2^{n/2})$，随机量子电路的谱间隙为 $\Omega(n^{-3})$。这些间隙在相应区域中与 $t$ 无关。进一步地，对于 $t \leq 2^{\Theta(n)}$，我们可将两个间隙改进为 $n^{-1}/\mathrm{polylog}(n, t)$，该结果直至多对数因子均为紧的。我们的谱间隙结果具有多重推论：1) 具有 $\mathcal{O}(n^4 t)$ 个门的随机可逆电路对所有 $t \geq 1$ 形成乘法误差的 $t$ 阶独立（偶）置换；对于 $t \leq \Theta(2^{n/6.1})$，我们证明 $\tilde{\mathcal{O}}(n^2 t)$ 个门即足够。2) 具有 $\mathcal{O}(n^4 t)$ 个门的随机量子电路对 $t \leq \Theta(2^{n/2})$ 形成乘法误差的酉 $t$-设计；对于 $t \leq \Theta(2^{2n/5})$，我们证明 $\tilde{\mathcal{O}}(n^2 t)$ 个门即足够。3) 随机电路的鲁棒量子电路复杂度在指数长时间内线性增长，从而证明了鲁棒 Brown–Susskind 猜想 [BS18, BCHJ+21]。我们的谱间隙界通过将随机量子电路约化为更具结构性的游走得以证明：该游走是 [MPSY24] 中“$\mathrm{PFC}$ 系综”的变体，结合了 Kassabov [Kas07a] 关于交替群的展开图（我们给出了其利用可逆电路的有效实现）。在约化过程中，我们利用无挫折哈密顿量研究中的工具，用局部随机电路近似结构化游走而不损失谱间隙。