Abstract：Matrix equations and matrix functions naturally produce operator outputs, making them important for quantum linear algebra beyond state preparation. We will present a sign-embedding framework that represents a range of such targets through the half-plane matrix sign of structured augmented matrices.

Combined with a logarithmic-sinc rational approximation, scaled multiplexing, and nodewise rebalancing, this approach gives block-encoding algorithms for Sylvester equations, generalized Lyapunov equations, matrix square roots and inverse square roots, matrix geometric means, and continuous-time algebraic Riccati equations. The framework applies beyond normal or diagonalizable inputs through field-of-values gap and strip-resolvent conditions, and provides a unified route to quantum algorithms for structured matrix problems.

Reference: Yanqiao Wang and Jin Peng Liu. Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions https://arxiv.org/abs/2604.25333

Bio: Yanqiao Wang is a Ph.D. student at Qiuzhen College, Tsinghua University, supervised by Prof. Jin-Peng Liu (YMSC) and Prof. Yang Liu (Institute for AI Industry Research).