No low-degree tests for quantum states

2026-07-31

AI Summary

This work addresses the problem of quantum low-degree testing: given an m-qudit quantum state, determine whether it is a phase state generated by a polynomial of degree at most d, or is far from all such states. The authors introduce a general analytical framework based on the classical decoding properties of dual codes and leverage bounds on the error tolerance of high-dimensional Reed–Muller codes under random noise. Using this approach, they establish—for the first time—a lower bound of Ω(⌊m/2⌋ choose ⌊(d−1)/2⌋) on the number of samples required by any quantum algorithm for this task. This result rules out the existence of efficient quantum low-degree testers and reveals a fundamental distinction between quantum and classical low-degree testing.

Abstract

We study the problem of testing low-degree phase states, namely m-qudit quantum states of the form $q^{-m/2} \sum_{x \in \mathbb{F}_q^m} ω^{f(x)} |x>$, where $f$ is a degree-$d$ polynomial. In contrast to the classical setting, where low-degree polynomials admit highly efficient classical testers, it is not known whether analogous quantum tests exist. We show that no such quantum low-degree test exists: any tester requires $Ω(\binom{\lfloor m/2\rfloor}{\lfloor (d-1)/2 \rfloor})$ copies to determine whether a given state is a degree-$d$ phase state or is far from every such state. Our results follow from a general framework that relates quantum testing of codeword phase states to classical decoding properties of the dual code, which allows us to leverage known bounds on the tolerance of high-rate Reed--Muller codes to random errors.

Problem

• quantum state testing
• low-degree phase states
• Reed-Muller codes
• quantum copy complexity

Innovation

• quantum low-degree testing
• phase states
• Reed–Muller codes
• quantum state testing
• dual code decoding