Contour-integral and Fourier transform based multivariable quantum eigenvalue transformation for commuting matrices
Abstract
We study the problem of implementing multivariable matrix-valued functions on a quantum computer and propose two quantum algorithms for multivariable matrix eigenvalue transformations acting on tuples of pairwise commuting matrices. The first algorithm is based on multivariable contour integrals and applies to arbitrary holomorphic functions on ℂn. The second algorithm is based on high-dimensional Fourier transforms and designed for smooth functions of commuting Hermitian matrices. For both algorithms, we discuss and analyze the complexity of their quantum implementation leveraging quantum singular value transformation, compression gadgets, and linear combination of unitaries. We additionally study their hybrid quantum--classical variants that reduce the number of required ancilla qubits to logarithmic dependence on the number of variables, at the cost of increased query complexity. As an application, we show that multivariable matrix polynomials can be implemented with no additional explicit degree dependence, depending only on global properties of the polynomials.
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BibTeX
@misc{jiang2026contourintegral,
title = {{Contour-integral and Fourier transform based multivariable quantum eigenvalue transformation for commuting matrices}},
author = {Shan Jiang and Dong An},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.32262},
eprint = {2609.32262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32262},
}
RIS
TY - UNPB
TI - Contour-integral and Fourier transform based multivariable quantum eigenvalue transformation for commuting matrices
AU - Shan Jiang
AU - Dong An
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2609.32262
UR - https://arxiv.org/abs/2609.32262
AB - We study the problem of implementing multivariable matrix-valued functions on a quantum computer and propose two quantum algorithms for multivariable matrix eigenvalue transformations acting on tuples of pairwise commuting matrices. The first algorithm is based on multivariable contour integrals and applies to arbitrary holomorphic functions on ℂ^n. The second algorithm is based on high-dimensional Fourier transforms and designed for smooth functions of commuting Hermitian matrices. For both algorithms, we discuss and analyze the complexity of their quantum implementation leveraging quantum singular value transformation, compression gadgets, and linear combination of unitaries. We additionally study their hybrid quantum--classical variants that reduce the number of required ancilla qubits to logarithmic dependence on the number of variables, at the cost of increased query complexity. As an application, we show that multivariable matrix polynomials can be implemented with no additional explicit degree dependence, depending only on global properties of the polynomials.
ER -