Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions

• URL: http://arxiv.org/abs/2604.25333v1
• Date: Tue, 28 Apr 2026 07:50:42 GMT
• Title: Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions
• Abstract summary: We develop a framework of operator-output quantum algorithms for matrix equations and matrix functions.<n>For ordinary Sylvester equations, we offer an explicit block-encoding of the target matrix solution.<n>These algorithms propagate the same overlap-based normalization bookkeeping to ordinary and generalized equations.
• Score: 4.591453488105558
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a systematic sign-embedding framework of operator-output quantum algorithms for matrix equations and matrix functions. Differing from the contour-integral treatment, we start with the matrix-sign embedding route: an augmented matrix $M$ whose half-plane matrix sign compresses the target operator either as a block of $\text{sign}(M)$ or, in projector form, through $(I-\text{sign}(M))/2$; we then construct a logarithmic-sinc approximation for the half-plane sign operator and combine it with structure-aware scaled multiplexing and nodewise rebalancing of shifted inverse families. For ordinary Sylvester equations, we offer an explicit block-encoding of the target matrix solution with query complexity linear in the inverse-conditioning parameters and logarithmic in the target error tolerance, under non-normal and non-diagonalizable settings given a field-of-values (FoV) gap or strip-resolvent hypotheses. These algorithms propagate the same overlap-based normalization bookkeeping to ordinary and generalized Sylvester equations, generalized Lyapunov equations, principal square roots and inverse square roots, matrix geometric means, and continuous-time algebraic Riccati equations (CARE). These results identify matrix-sign embeddings and nodewise rebalancing as reusable design principles for structured operator-output quantum linear algebra.

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