How to efficiently encode classical data is a fundamental task in quantum computing. While many existing works treat classical data encoding as a black box in oracle-based quantum algorithms, their explicit constructions are crucial for the efficiency of practical algorithm implementations. Here, we unveil the mystery of the classical data encoding black box and study the Clifford + T complexity in constructing several typical quantum access models. For general matrices (even including sparse ones), we prove that sparse-access input models and block-encoding both require nearly linear circuit complexities relative to the matrix dimension. We also give construction protocols achieving near-optimal gate complexities. On the other hand, the construction becomes efficient with respect to the data qubit when the matrix is a linear combination of polynomial terms of efficiently implementable unitaries. As a typical example, we propose improved block-encoding when these unitaries are Pauli strings. Our protocols are built upon improved quantum state preparation and a select oracle for Pauli strings, which hold independent values. Our access model constructions provide considerable flexibility, allowing for tunable ancillary qubit numbers and offering corresponding space-time trade-offs.

如何有效地编码经典数据是量子计算的一项基本任务。虽然许多现有的工作将经典数据编码视为基于预言的量子算法中的黑匣子，但它们的显式构造对于实际算法实现的效率至关重要。在这里，我们揭开了经典数据编码黑匣子的神秘面纱，并研究了 构建几种典型量子访问模型的Clifford + T复杂度。对于一般矩阵（甚至包括稀疏矩阵），我们证明稀疏访问输入模型和块编码都需要相对于矩阵维度的近线性电路复杂性。我们还提供了实现接近最佳门复杂性的构建协议。另一方面，当矩阵是可有效实现酉式的多项式项的线性组合时，对于数据量子位而言，构造变得高效。作为一个典型的例子，当这些酉元是泡利字符串时，我们提出改进的块编码。我们的协议建立在改进的量子态准备和保利弦的精选预言机之上，保利弦具有独立的值。我们的访问模型结构提供了相当大的灵活性，允许可调的辅助量子比特数并提供相应的时空权衡。