标题： Clifford 群构成一个幺正 3-设计 显示英文标题

标题： The Clifford group forms a unitary 3-design

摘要： 酉$k$-设计是一组有限的酉矩阵集合，它们近似于酉矩阵上的 Haar 分布。 已知若干个集合是 2-设计，包括 Clifford 群上的均匀分布，但之前没有任何族被证明是 3-设计。 我们证明了 Clifford 群是 3-设计，表明它比以前预期的更好地逼近 Haar 随机酉矩阵。 我们的证明策略适用于满足我们称为 Pauli 2-混合性质的任意酉分布，并且无需使用繁重的数学工具。 我们还证明了 Clifford 群不是 4-设计，从而刻画了随机 Clifford 元素逼近 Haar 随机酉矩阵的程度。 此外，我们证明了对于量子比特的广义 Clifford 群，如果量子比特的维度不是 2 的幂，则它不是 3-设计。 显示英文摘要

摘要： Unitary $k$-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machinery. We also show that the Clifford group does not form a 4-design, thus characterizing how well random Clifford elements approximate Haar-random unitaries. Additionally, we show that the generalized Clifford group for qudits is not a 3-design unless the dimension of the qudit is a power of 2.