aiqbench analysis
This research advances quantum numerical linear algebra by introducing Fourier extensions to decouple approximation accuracy from subnormalization in block encodings. By optimizing redundant coefficients for Hermitian matrix inputs, the method achieves exponential convergence rates while maintaining error independence, which is critical for scaling algorithms like the Quantum Linear Systems Algorithm (QLSA).
Fourier extensions for matrix-function block encodings with error-independent subnormalization bounds
arXiv:2601.18024v2 Announce Type: replace Abstract: Block encodings of non-unitary matrix functions are central to quantum numerical linear algebra. Hamiltonian simulation is a natural input model for Hermitian matrices, but accurate block encodings often incur large subnormalization. We decouple the accuracy from the subnormalization by formulating the matrix-function block encoding as a Fourier-extension approximation problem, yielding a linear combination of unitaries for Hermitian matrix inputs. Fourier extensions approximate non-periodic functions by a Fourier series on a larger periodic domain, creating redundant coefficients that can be optimized for their absolute sum, and hence subnormalization. The coefficients may be chosen for optimal subnormalization with algebraic convergence, by tuning the subnormalization bound for increasing rates of exponential convergence, or by Sobolev-regularized fitting to accommodate more general spectral sets. Fourier-extension block encodings apply to eigenvalue transforms of Hermitian matrices, or to odd singular-value transforms of general matrices, including as a quantum linear systems algorithm.
Abstract:Block encodings of non-unitary matrix functions are central to quantum numerical linear algebra. Hamiltonian simulation is a natural input model for Hermitian matrices, but accurate block encodings often incur large subnormalization. We decouple the accuracy from the subnormalization by formulating the matrix-function block encoding as a Fourier-extension approximation problem, yielding a linear combination of unitaries for Hermitian matrix inputs. Fourier extensions approximate non-periodic functions by a Fourier series on a larger periodic domain, creating redundant coefficients that can be optimized for their absolute sum, and hence subnormalization. The coefficients may be chosen for optimal subnormalization with algebraic convergence, by tuning the subnormalization bound for increasing rates of exponential convergence, or by Sobolev-regularized fitting to accommodate more general spectral sets. Fourier-extension block encodings apply to eigenvalue transforms of Hermitian matrices, or to odd singular-value transforms of general matrices, including as a quantum linear systems algorithm.
arXiv:2601.18024 [quant-ph]
(or arXiv:2601.18024v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2601.18024 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 233 words
