Random circuits giving rise to unitary designs are key tools in quantum information science and many-body physics. In this work, we investigate a class of random quantum circuits with a specific gate structure. Within this framework, we prove that one-dimensional structured random circuits with non-Haar random local gates can exhibit substantially more global randomness compared to Haar random circuits with the same underlying circuit architecture. In particular, we derive all the exact eigenvalues and eigenvectors of the second-moment operators for these structured random circuits under a solvable condition, by establishing a link to the Kitaev chain, and show that their spectral gaps can exceed those of Haar random circuits. Our findings have applications in improving circuit depth bounds for randomized benchmarking and the generation of approximate unitary 2-designs from shallow random circuits.

翻译：产生酉设计的随机电路是量子信息科学与多体物理中的关键工具。本研究探讨一类具有特定门结构的随机量子电路。在此框架下，我们证明一维结构化随机电路即使采用非哈尔随机局部门，也能展现出比相同电路架构下哈尔随机电路显著更强的全局随机性。特别地，通过建立与Kitaev链的联系，我们在可解条件下推导出这些结构化随机电路二阶矩算子的所有精确特征值与特征向量，并证明其谱隙可超越哈尔随机电路。我们的发现对改进随机基准测试的电路深度界限以及从浅层随机电路生成近似酉2-设计具有应用价值。