Fourier Analysis on Quantum Systems and Applications

University Of South Carolina At Columbia, Columbia SC

Investigators

Abstract

Fourier analysis is a foundational area of mathematics with widespread applications. One of its key ideas is to understand a signal through both its time-domain and frequency-domain representations. In theoretical computer science and learning theory, signals often arise in discrete settings with rich structure—for example, as functions on the Hamming cube (i.e., the space of binary strings). Such functions often exhibit low complexity when analyzed through their Fourier expansion or frequency components. Tackling questions in these domains requires a deep understanding of these structured, low-complexity functions, for which Fourier analytic tools have proven useful. As quantum computing rapidly advances, the natural setting shifts from classical bits to qubits, where signals are represented not by functions but by operators that are non-commutative in nature. This introduces new challenges that demand extensions of classical Fourier analysis tools to the quantum setting. Addressing these challenges is the goal of this project. The project will include efforts in training both undergraduate and graduate students through PI’s mentoring role and in serving the research community through organizing conferences and workshops. This study investigates quantum analogues of low-complexity functions on the Hamming cube, where complexity is measured, for example, by influence, degree, or the number of variables on which a function depends. A central challenge in this context arises when the dimension tends to infinity. Although Fourier analytic tools have proven highly effective for such problems in the classical setting—illustrated by foundational results on Boolean functions by Kahn, Kalai, and Linial, as well as Talagrand, and by dimension-free inequalities originating from the early work of Littlewood, Bohnenblust and Hille—they do not readily extend to the quantum realm due to inherent non-commutativity and the absence of an underlying classical space. The principal investigator will therefore develop tools to address challenges from the high-dimensional regime and non-commutativity, thereby strengthening the connections between harmonic analysis, functional analysis, quantum information theory, and related areas. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.