Quantum oblique eigenprojection

Abstract:Every square matrix decomposes its underlying Hilbert space into generalized, nonorthogonal eigensubspaces. We show that a quantum computer can perform such an oblique eigenprojection $\Pi$ given block encoding access to the input matrix. Our approach has a query complexity nearly linear in the inverse gap and a normalization factor close to $\lVert\Pi\rVert$ under a spectral-set condition on the input. This covers common assumptions on the numerical range or diagonalizability and matches known results for orthogonal eigenprojections. We achieve this with a two-sided block preconditioning that uses a discrete Fourier transform of the matrix resolvent. We describe applications to: (i) preparing eigenstates of matrices with complex eigenvalues, extending the quantum eigenvalue transformation algorithm of Low and Su beyond real spectra; (ii) solving continuous-time algebraic Riccati equations, cubically speeding up a prior solver of Rodenas-Ruiz, Zhao, and Lee; and (iii) solving ordinary Sylvester equations, quadratically improving a direct augmented method of Wang and Liu. Our result suggests a promising route to applying nonanalytic matrix functions on quantum computers.

Quantum Physics (quant-ph); Data Structures and Algorithms (cs.DS); Chemical Physics (physics.chem-ph)

arXiv:2609.40254 [quant-ph]

(or arXiv:2609.40254v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.40254 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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