Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions

TL;DR

This paper introduces a sign-embedding framework for quantum algorithms solving matrix equations and functions, offering efficient solutions under certain spectral conditions.

Contribution

It develops a systematic sign-embedding approach for operator-output quantum algorithms, extending to various matrix equations and functions with improved complexity.

Findings

Explicit block-encoding for Sylvester solutions with query complexity linear in inverse-conditioning

Applicable to non-normal, non-diagonalizable matrices with spectral gap assumptions

Unified approach for multiple matrix functions and equations using sign-embedding principles

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.