Quantum Higher Order Fourier Analysis and the Clifford Hierarchy
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918136657215488 |
|---|---|
| author | Bu, Kaifeng Gu, Weichen Jaffe, Arthur |
| author_facet | Bu, Kaifeng Gu, Weichen Jaffe, Arthur |
| contents | We propose a mathematical framework that we call quantum, higher-order Fourier analysis. This generalizes the classical theory of higher-order Fourier analysis, which led to many advances in number theory and combinatorics. We define a family of quantum measures on a Hilbert space, that reduce in the case of diagonal matrices to the classical uniformity norms. We show that our quantum measures and our related theory of quantum higher-order Fourier analysis characterize the Clifford hierarchy, an important notion of complexity in quantum information. In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the k-th level of the Clifford hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15908 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum Higher Order Fourier Analysis and the Clifford Hierarchy Bu, Kaifeng Gu, Weichen Jaffe, Arthur Quantum Physics Mathematical Physics Functional Analysis Operator Algebras We propose a mathematical framework that we call quantum, higher-order Fourier analysis. This generalizes the classical theory of higher-order Fourier analysis, which led to many advances in number theory and combinatorics. We define a family of quantum measures on a Hilbert space, that reduce in the case of diagonal matrices to the classical uniformity norms. We show that our quantum measures and our related theory of quantum higher-order Fourier analysis characterize the Clifford hierarchy, an important notion of complexity in quantum information. In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the k-th level of the Clifford hierarchy. |
| title | Quantum Higher Order Fourier Analysis and the Clifford Hierarchy |
| topic | Quantum Physics Mathematical Physics Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2508.15908 |