Quantum and Classical Low-Degree Learning via a Dimension-Free Remez Inequality

Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including to the slice binom([n],k), the hypergrid [K]ⁿ, and noncommutative spaces (matrix algebras). We present here a new way to relate functions on the hypergrid (or products of cyclic groups) to their harmonic extensions over the polytorus. We show the supremum of a function f over products of the cyclic group {exp(2π i k/K)}_{k = 1}^K controls the supremum of f over the entire polytorus ({z ∈ ℂ:|z| = 1}ⁿ), with multiplicative constant C depending on K and deg(f) only. This Remez-type inequality appears to be the first such estimate that is dimension-free (i.e., C does not depend on n). This dimension-free Remez-type inequality removes the main technical barrier to giving 𝒪(log n) sample complexity, polytime algorithms for learning low-degree polynomials on the hypergrid and low-degree observables on level-K qudit systems. In particular, our dimension-free Remez inequality implies new Bohnenblust-Hille-type estimates which are central to the learning algorithms and appear unobtainable via standard techniques. Thus we extend to new spaces a recent line of work [Eskenazis and Ivanisvili, 2022; Huang et al., 2022; Volberg and Zhang, 2023] that gave similarly efficient methods for learning low-degree polynomials on the hypercube and observables on qubits. An additional product of these efforts is a new class of distributions over which arbitrary quantum observables are well-approximated by their low-degree truncations - a phenomenon that greatly extends the reach of low-degree learning in quantum science [Huang et al., 2022].