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 [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(2 ik/K)}Kk =1 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 [10, 13, 23] 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 [13].