AI 中文总结
该研究针对量子态的低度相位态测试问题，证明不存在对应的量子低度测试，需特定数量的量子副本，其结果基于关联量子测试与对偶码译码特性的通用框架，利用了Reed–Muller码的已知容限界。
AI 中文摘要
我们研究低度相位态的测试问题，即形如 $q^{-m/2} \sum_{x \in \mathbb{F}_q^m} \omega^{f(x)} |x\rangle$ 的 $m$ 量子位量子态，其中 $f$ 是 $d$ 次多项式。与经典场景中低度多项式存在高效经典测试器不同，目前尚不清楚是否存在类似的量子测试器。我们证明不存在此类量子低度测试：任何测试器都需要 $\Omega(\binom{\lfloor m/2\rfloor}{\lfloor (d-1)/2 \rfloor})$ 个副本，才能判断给定态是 $d$ 次相位态还是与所有此类态相距甚远。我们的结果基于一个通用框架，该框架将码字相位态的量子测试与对偶码的经典译码特性关联起来，使我们能够利用已知的高码率Reed–Muller码对随机错误的容限界。
英文摘要
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.
Comments37 pages
