In this work we give an efficient construction of unitary $k$-designs using $\tilde{O}(k\cdot poly(n))$ quantum gates, as well as an efficient construction of a parallel-secure pseudorandom unitary (PRU). Both results are obtained by giving an efficient quantum algorithm that lifts random permutations over $S(N)$ to random unitaries over $U(N)$ for $N=2^n$. In particular, we show that products of exponentiated sums of $S(N)$ permutations with random phases approximately match the first $2^{\Omega(n)}$ moments of the Haar measure. By substituting either $\tilde{O}(k)$-wise independent permutations, or quantum-secure pseudorandom permutations (PRPs) in place of the random permutations, we obtain the above results. The heart of our proof is a conceptual connection between the large dimension (large-$N$) expansion in random matrix theory and the polynomial method, which allows us to prove query lower bounds at finite-$N$ by interpolating from the much simpler large-$N$ limit. The key technical step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-$N$ expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension $N$.

翻译：本文给出一种利用 $\tilde{O}(k\cdot poly(n))$ 个量子门高效构造酉 $k$-设计的方法，以及并行安全伪随机酉（PRU）的高效构造。两结果均通过给出将 $S(N)$ 上随机排列提升至 $U(N)$（$N=2^n$）上随机酉的高效量子算法实现。特别地，我们证明带随机相位的 $S(N)$ 排列的指数和乘积可近似匹配 Haar 测度的前 $2^{\Omega(n)}$ 阶矩。通过将随机排列替换为 $\tilde{O}(k)$ 阶独立排列或量子安全伪随机排列（PRP），即可获得上述结果。证明核心在于随机矩阵理论中大维度（大 $N$）展开与多项式方法之间的概念联系——这使得我们能够通过从更简单的大 $N$ 极限插值，在有限 $N$ 情形下证明查询下界。关键技术步骤是构建具有低阶大 $N$ 展开性质的分割代数不可约表示的标准正交基，从而证明区分概率是维度 $N$ 的低阶有理多项式。