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：区块编码是许多现有量子算法的核心。 同时, 稠密操作员的有效和明确的区块编码被公认为是一个具有挑战性的问题。 本文件全面研究了密密操作员大家庭的区块编码: 假相操作员( PDOs) 。 首先, 开发了一个通用 PDOs 的区块编码办法。 然后, 我们为具有可分离结构的 PDOs 提出了一个效率更高的方案。 最后, 我们为PDOs 展示了一种明确有效的区块编码算法, 其尺寸与维度完全可分离的结构。 对所有区块编码算法都进行了复杂的分析。 理论结果的应用以工作实例来说明, 包括可变系数椭圆形操作员的表示, 以及在不引用量子线性算法( QLSAs) 的情况下计算椭圆形操作员的逆函数 。