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 aris