Quantum Algorithms for Gowers Norm Estimation, Polynomial Testing, and Arithmetic Progression Counting over Finite Abelian Groups

TL;DR

This paper introduces quantum algorithms for estimating Gowers norms over finite abelian groups, enabling efficient detection of algebraic structures, polynomial testing, and arithmetic progression counting, with applications in property testing and noise resilience.

Contribution

The work generalizes quantum Gowers norm estimation to arbitrary finite abelian groups and higher orders, providing new algorithms for polynomial structure testing and progressions counting.

Findings

Quantum algorithms estimate Gowers norms efficiently over finite abelian groups.

Quasipolynomial-time quantum algorithms distinguish degree-d phase polynomials.

The methods are resilient to certain quantum noise models, suitable for NISQ devices.

Abstract

We propose a family of quantum algorithms for estimating Gowers uniformity norms $ U^k $ over finite abelian groups and demonstrate their applications to testing polynomial structure and counting arithmetic progressions. Building on recent work for estimating the $ U^2 $-norm over $ \mathbb{F}_2^n $, we generalize the construction to arbitrary finite fields and abelian groups for higher values of $ k $. Our algorithms prepare quantum states encoding finite differences and apply Fourier sampling to estimate uniformity norms, enabling efficient detection of structural correlations. As a key application, we show that for certain degrees $ d = 4, 5, 6 $ and under appropriate conditions on the underlying field, there exist quasipolynomial-time quantum algorithms that distinguish whether a bounded function $ f(x) $ is a degree-$ d $ phase polynomial or far from any such structure. These algorithms leverage recent inverse theorems for Gowers norms, together with amplitude estimation, to reveal higher-order algebraic correlations. We also develop a quantum method for estimating the number of 3-term arithmetic progressions in Boolean functions $ f : \mathbb{F}_p^n \to \{0,1\} $, based on estimating the $ U^2 $-norm. Though not as query-efficient as Grover-based counting, our approach provides a structure-sensitive alternative aligned with additive combinatorics. Finally, we demonstrate that our techniques remain valid under certain quantum noise models, due to the shift-invariance of Gowers norms. This enables noise-resilient implementations within the NISQ regime and suggests that Gowers-norm-based quantum algorithms may serve as robust primitives for quantum property testing, learning, and pseudorandomness.