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 $Z(\mathbb{C}[G])$, 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.

Quantum Physics (quant-ph); Representation Theory (math.RT)

arXiv:2610.16827 [quant-ph]

(or arXiv:2610.16827v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.16827 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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