Block encoding lies at the core of many existing quantum algorithms. Meanwhile, efficient and explicit block encodings of dense operators are commonly acknowledged as a challenging problem. This paper presents a comprehensive study of the block encoding of a rich family of dense operators: the pseudo-differential operators (PDOs). First, a block encoding scheme for generic PDOs is developed. Then we propose a more efficient scheme for PDOs with a separable structure. Finally, we demonstrate an explicit and efficient block encoding algorithm for PDOs with a dimension-wise fully separable structure. Complexity analysis is provided for all block encoding algorithms presented. The application of theoretical results is illustrated with worked examples, including the representation of variable coefficient elliptic operators and the computation of the inverse of elliptic operators without invoking quantum linear system algorithms (QLSAs).

翻译：块编码是许多现有量子算法的核心。与此同时，稠密算子的有效且显式块编码被公认为一个具有挑战性的问题。本文对一类丰富的稠密算子——拟微分算子（PDOs）的块编码进行了全面研究。首先，开发了一种针对一般PDOs的块编码方案。接着，我们针对具有可分离结构的PDOs提出了一种更高效的方案。最后，我们展示了一种针对具有维度全可分离结构的PDOs的显式且高效的块编码算法。文中对所提出的所有块编码算法均提供了复杂度分析。理论结果的应用通过具体实例加以说明，包括变系数椭圆算子的表示以及在不调用量子线性系统算法（QLSAs）的情况下计算椭圆算子逆的方法。