Unitary $T$-designs play an important role in quantum information, with diverse applications in quantum algorithms, benchmarking, tomography, and communication. Until now, the most efficient construction of unitary $T$-designs for $n$-qudit systems has been via random local quantum circuits, which have been shown to converge to approximate $T$-designs in the diamond norm using $O(T^{5+o(1)} n^2)$ quantum gates. In this work, we provide a new construction of $T$-designs via random matrix theory using $\tilde{O}(T^2 n^2)$ quantum gates. Our construction leverages two key ideas. First, in the spirit of central limit theorems, we approximate the Gaussian Unitary Ensemble (GUE) by an i.i.d. sum of random Hermitian matrices. Second, we show that the product of just two exponentiated GUE matrices is already approximately Haar random. Thus, multiplying two exponentiated sums over rather simple random matrices yields a unitary $T$-design, via Hamiltonian simulation. A central feature of our proof is a new connection between the polynomial method in quantum query complexity and the large-dimension ($N$) expansion in random matrix theory. In particular, we show that the polynomial method provides exponentially improved bounds on the high moments of certain random matrix ensembles, without requiring intricate Weingarten calculations. In doing so, we define and solve a new type of moment problem on the unit circle, asking whether a finite number of equally weighted points, corresponding to eigenvalues of unitary matrices, can reproduce a given set of moments.

翻译：酉$T$-设计在量子信息中扮演重要角色，广泛应用于量子算法、基准测试、层析成像和通信等领域。迄今为止，针对$n$量子位系统最有效的酉$T$-设计构造是通过随机局域量子电路实现的，此类电路已被证明在金刚石范数下使用$O(T^{5+o(1)} n^2)$个量子门可收敛至近似$T$-设计。本文通过随机矩阵理论提出一种新的$T$-设计构造方法，仅需$\tilde{O}(T^2 n^2)$个量子门。我们的构造基于两个关键思想：其一，遵循中心极限定理的精神，利用独立同分布随机厄米矩阵之和来近似高斯酉系综（GUE）；其二，证明仅两个指数化GUE矩阵的乘积已接近哈尔随机分布。因此，通过哈密顿模拟将两个指数化简单随机矩阵求和结果相乘，即可获得酉$T$-设计。我们证明的核心特征在于建立了量子查询复杂度中的多项式方法与随机矩阵理论中大维度（$N$）展开之间的新联系。特别地，我们证明多项式方法能对特定随机矩阵系综的高阶矩给出指数级改进的界，而无需复杂的Weingarten计算。在此过程中，我们在单位圆上定义并求解了一种新型矩问题——判断有限个等权重点（对应于酉矩阵的特征值）能否再现给定矩集合。