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arXiv · 2610.16827

The Quantum Character Transform

Kagwe A. Muchane
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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 $Z(\mathbb{C}[G])$, 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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Kagwe A. Muchane. 2026-10-12. The Quantum Character Transform. https://arxiv.org/abs/2610.16827

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