Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions

Abstract

We develop operator-output quantum algorithms for matrix equations and matrix functions using matrix-sign embeddings. For each problem, an augmented matrix $M$ is chosen so that the target operator is a block of $\text{sign}(M)$ or is recovered from blocks of $(I-\text{sign}(M))/2$. We approximate the half-plane sign by a logarithmic-sinc formula and implement the shifted inverse families by scaled multiplexing with node-dependent rebalancing. For ordinary Sylvester equations, this yields a block-encoding of the solution with query complexity linear in the relevant conditioning parameters and logarithmic in the inverse error tolerance, under either a field-of-values (FoV) gap or a strip-resolvent bound. The same method extends to generalized Sylvester and Lyapunov equations, principal square and inverse square roots, matrix geometric means, and continuous-time algebraic Riccati equations (CARE), with explicit query-complexity and block-encoding-normalization bounds that cover non-normal and non-diagonalizable cases.