Quantum and classical low-degree learning via a dimension-free Remez inequality

TL;DR

This paper introduces a dimension-free Remez inequality for functions on the hypergrid, enabling efficient learning of low-degree polynomials and quantum observables across various spaces, with implications for quantum science and algorithm design.

Contribution

It establishes the first dimension-free Remez inequality for functions on the hypergrid, facilitating improved learning algorithms for low-degree polynomials and quantum observables.

Findings

Dimension-free Remez inequality for hypergrid functions.

Enhanced sample complexity and polynomial learning algorithms.

New bounds for quantum observable approximation.

Abstract

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}.