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 $\{\exp(2\pi i k/K)\}_{k=1}^K$ controls the supremum of $f$ over the entire polytorus $(\{z\in\mathbf{C}:|z|=1\}^n)$, with multiplicative constant $C$ depending on $K$ and $\text{deg}(f)$ 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 $\mathcal{O}(\log n)$ 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 \cite{EI22, CHP, VZ22} 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 \cite{CHP}.