Quantum Talagrand, KKL and Friedgut's theorems and the learnability of quantum Boolean functions
Cambyse Rouzé, Melchior Wirth, Haonan Zhang
TL;DR
The paper advances quantum Boolean analysis by establishing quantum analogues of the classic KKL, Friedgut Junta, and Talagrand results using noncommutative hypercontractivity and gradient estimates. It proves a quantum $L^1$-Poincaré inequality, a quantum $L^1$-Talagrand inequality, a KKL-type bound for $L^1$-influences, and a Friedgut Junta theorem with explicit dependency on quantum influences, then generalizes these results to abstract von Neumann algebras. Through concrete examples (classical, generalized depolarizing, quantum Ornstein–Uhlenbeck, and group von Neumann algebras), the work demonstrates broad applicability, and develops implications for quantum circuit complexity, learning of quantum observables and dynamics, and quantum isoperimetry. Overall, the results provide a robust noncommutative framework for influence, concentration, and learnability in quantum information science, with potential extensions to continuous variables and noncommutative geometry.
Abstract
We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL Theorem, Friedgut's Junta Theorem and Talagrand's variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables.