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：本文提出一种使用$\tilde{O}(k\cdot poly(n))$量子门的高效酉$k$-设计构造方法，以及并行安全的伪随机酉矩阵（PRU）的高效构造方案。两项成果均通过构建高效量子算法实现，该算法可将$S(N)$上的随机置换提升至$U(N)$上的随机酉矩阵，其中$N=2^n。特别地，我们证明：具有随机相位的$S(N)$置换指数和之乘积，能以近似方式匹配哈尔测度的前$2^{\Omega(n)}$阶矩。通过采用$\tilde{O}(k)$-wise独立置换或量子安全伪随机置换（PRP）替代随机置换，即可获得上述结果。证明的核心在于建立随机矩阵理论中大维度（大$N$）展开与多项式方法的概念关联，该关联使我们能够通过从更简单的大$N$极限进行插值，证明有限$N$情形下的查询下界。关键技术步骤是展示分割代数不可约表示的一组具有低阶大$N$展开的正交基，由此可证明区分概率是维度$N$的低阶有理多项式。