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 Π 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 lVertΠrVert 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.
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BibTeX
@misc{dalzell2026quantum,
title = {{Quantum oblique eigenprojection}},
author = {Alexander M. Dalzell and Yuan Su},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40254},
eprint = {2609.40254},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40254},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Quantum oblique eigenprojection
AU - Alexander M. Dalzell
AU - Yuan Su
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40254
UR - https://arxiv.org/abs/2609.40254
AB - Every square matrix decomposes its underlying Hilbert space into generalized, nonorthogonal eigensubspaces. We show that a quantum computer can perform such an oblique eigenprojection Π 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 lVertΠrVert 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.
ER -