Quantum higher-order Fourier analysis and the Clifford hierarchy.
Where this comes from
- Record sourced from PubMed, PMID 41201827.
- Also identified by DOI 10.1073/pnas.2515667122 and PMC identifier 12625977.
- Licence recorded as CC BY.
- The licence permits redistribution, so the abstract is shown in full and the full text is available from the publisher.
Abstract
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 recent advances in number theory and combinatorics. We define a family of "quantum measures" on linear transformations on a Hilbert space, that reduce in the case of diagonal matrices to the uniformity norms introduced by Timothy Gowers. 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 computation. In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the [Formula: see text] level of the Clifford hierarchy.