AI 中文总结
该研究针对仅用系统量子比特在对数深度内生成强酉设计的开放性问题，构造出最优深度的强近似酉k-设计，通过新的Pauli混合界结合现有结果解决了该问题。
AI 中文摘要
酉设计提供了哈尔随机酉算子的有限矩近似，广泛应用于物理学和量子信息领域，涵盖 scrambling、黑洞动力学以及量子算法的基础 primitives。强酉设计对应更严格的操作近似概念，要求即使量子算法可正向访问酉算子，也可通过其逆、转置和复共轭访问时，仍无法将其与哈尔随机性区分。受限于 scrambling 需在系统内部产生的物理要求，Schuster、Ma、Lombardi、Brandão 和 Huang（arXiv:2509.26310）提出一个开放性问题：能否仅用系统量子比特在对数深度内生成强酉设计。对于每个固定的设计阶数k和可测量误差容限，我们仅用n个原始系统量子比特，在最优Θ(log n)全对全电路深度内构造出强近似酉k-设计。我们的新要素是完美匹配系综的对数深度Pauli混合界，其层将量子比特均匀随机配对并应用独立随机两量子比特门，该界可控制混合正反向双查询情形，结合现有设计与拼接结果，得到任意固定阶数的强酉设计。
英文摘要
Unitary designs provide finite-moment approximations to Haar-random unitaries, with wide-ranging applications across physics and quantum information, from scrambling and black-hole dynamics to foundational primitives in quantum algorithms. Strong unitary designs capture a more demanding operational notion of approximation, requiring indistinguishability from Haar randomness even for quantum algorithms that may access a unitary not only in the forward direction, but also through its inverse, transpose, and complex conjugate. Motivated by the physical requirement that scrambling arise within the system itself, Schuster, Ma, Lombardi, Brandão, and Huang (arXiv:2509.26310) left open whether strong unitary designs can be generated in logarithmic depth using only the system qubits. For every fixed design order $k$ and measurable-error tolerance, we construct strong approximate unitary $k$-designs in optimal $Θ(\log n)$ all-to-all circuit depth using only the $n$ original system qubits. Our new ingredient is a logarithmic-depth Pauli-mixing bound for the perfect-matching ensemble, whose layers pair the qubits uniformly at random and apply independent random two-qubit gates. This bound controls the mixed forward-reverse two-query case, which we combine with existing design and gluing results to obtain strong unitary designs of arbitrary fixed order.
