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Updated: Jan 6, 2026
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
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Quantum higher-order Fourier analysis and the Clifford hierarchy
Kaifeng Bu1,2, Weichen Gu1,3, Arthur Jaffe2,4
1Department of Mathematics, The Ohio State University, Columbus, OH 43210.
Summary
We introduce quantum higher-order Fourier analysis, a new mathematical framework generalizing classical methods. This framework defines quantum measures that characterize the Clifford hierarchy in quantum computation.
Area of Science:
- Quantum Information Theory
- Mathematical Physics
- Harmonic Analysis
Background:
- Classical higher-order Fourier analysis has driven advances in number theory and combinatorics.
- The Clifford hierarchy is a crucial concept for understanding quantum computation complexity.
Purpose of the Study:
- To develop a mathematical framework for quantum higher-order Fourier analysis.
- To define quantum measures generalizing classical uniformity norms.
- To establish a connection between quantum measures and the Clifford hierarchy.
Main Methods:
- Development of a novel mathematical framework for quantum higher-order Fourier analysis.
- Definition of a family of quantum measures on linear transformations in Hilbert spaces.
- Analysis of the relationship between quantum measures and the Clifford hierarchy.
Main Results:
- Quantum measures are introduced, generalizing classical uniformity norms for diagonal matrices.
- The proposed framework is shown to characterize the Clifford hierarchy.
- A necessary and sufficient analytic condition is derived for unitaries belonging to specific levels of the Clifford hierarchy.
Conclusions:
- Quantum higher-order Fourier analysis provides a powerful tool for studying quantum computation.
- The defined quantum measures offer new insights into the structure of the Clifford hierarchy.
- This work establishes a significant link between harmonic analysis and quantum complexity theory.
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