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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2604.25333 |
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| _version_ | 1866914513641537536 |
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| author | Wang, Yanqiao Liu, Jin-Peng |
| author_facet | Wang, Yanqiao Liu, Jin-Peng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25333 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions Wang, Yanqiao Liu, Jin-Peng Quantum Physics Numerical Analysis 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. |
| title | Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions |
| topic | Quantum Physics Numerical Analysis |
| url | https://arxiv.org/abs/2604.25333 |