The Quantum Character Transform
Abstract
Bastidas et al. (2025) highlighted the importance of exploiting group symmetries to reduce computational overhead and unify simulation methods across both condensed matter physics and ab initio quantum chemistry. I discuss the discrete character transform, which is the group Fourier transform restricted to Z(ℂ[G]), the center of the group algebra, where class functions form a commutative convolution algebra and irreducible characters are its spectral basis. The normalized character table of a non-abelian group can be obtained through a restriction operator, which means an efficient quantum character transform for any finite group with an efficient quantum Fourier transform. I give complementary quantum algorithms for class convolution based on direct class-sum block encoding and spectral implementation in the character basis, explaining their applications to symmetry-adapted many-body dynamics, molecular and spectroscopic symmetry analysis, and quantum walks, as well as characterizing the central information used in hidden-subgroup problems, including its limitations for graph isomorphism.
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BibTeX
@misc{muchane2026quantum,
title = {{The Quantum Character Transform}},
author = {Kagwe A. Muchane},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16827},
eprint = {2610.16827},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16827},
}
RIS
TY - UNPB
TI - The Quantum Character Transform
AU - Kagwe A. Muchane
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16827
UR - https://arxiv.org/abs/2610.16827
AB - Bastidas et al. (2025) highlighted the importance of exploiting group symmetries to reduce computational overhead and unify simulation methods across both condensed matter physics and ab initio quantum chemistry. I discuss the discrete character transform, which is the group Fourier transform restricted to Z(ℂ[G]), the center of the group algebra, where class functions form a commutative convolution algebra and irreducible characters are its spectral basis. The normalized character table of a non-abelian group can be obtained through a restriction operator, which means an efficient quantum character transform for any finite group with an efficient quantum Fourier transform. I give complementary quantum algorithms for class convolution based on direct class-sum block encoding and spectral implementation in the character basis, explaining their applications to symmetry-adapted many-body dynamics, molecular and spectroscopic symmetry analysis, and quantum walks, as well as characterizing the central information used in hidden-subgroup problems, including its limitations for graph isomorphism.
ER -