On efficient quantum block encoding of pseudo-differential operators

TL;DR

This paper develops efficient quantum block encoding schemes for pseudo-differential operators, including separable and fully separable structures, enabling improved quantum algorithms for PDEs without relying on quantum linear system algorithms.

Contribution

It introduces new quantum block encoding methods for PDOs, especially for separable and fully separable cases, with explicit algorithms and complexity analysis.

Findings

Efficient block encoding schemes for generic PDOs.

Specialized encoding algorithms for separable PDOs.

Application examples including elliptic operators and their inverses.

Abstract

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).