标题： Clifford群形成一个酉3设计 显示英文标题

标题： The Clifford group forms a unitary 3-design

摘要： 酉 $k$-设计是有限的酉矩阵集合，它们近似酉矩阵上的哈尔分布。 已知几个集合是精确的2-设计，包括克利福德群上的均匀分布，但尚未发现任何数量的量子比特的精确3-设计的集合族。 我们证明了克利福德群是一个3-设计，表明它比以前预期的更接近哈尔随机酉矩阵。 我们的证明策略适用于满足我们称为泡利2混合性质的任何酉分布，并且无需使用复杂的数学工具。 我们还表明，克利福德群不是一个4-设计，从而确定了随机克利福德元素近似哈尔随机酉矩阵的程度。 此外，我们表明，维度大于2的量子系统的广义克利福德群不是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 exact 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles is known to form an exact 3-design for any number of qubits. 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 of dimension greater than 2 is not a 3-design.