We consider the problem of learning low-degree quantum objects up to $\varepsilon$-error in $\ell_2$-distance. We show the following results: $(i)$ unknown $n$-qubit degree-$d$ (in the Pauli basis) quantum channels and unitaries can be learned using $O(1/\varepsilon^d)$ queries (independent of $n$), $(ii)$ polynomials $p:\{-1,1\}^n\rightarrow [-1,1]$ arising from $d$-query quantum algorithms can be classically learned from $O((1/\varepsilon)^d\cdot \log n)$ many random examples $(x,p(x))$ (which implies learnability even for $d=O(\log n)$), and $(iii)$ degree-$d$ polynomials $p:\{-1,1\}^n\to [-1,1]$ can be learned through $O(1/\varepsilon^d)$ queries to a quantum unitary $U_p$ that block-encodes $p$. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded~polynomials.

翻译：我们考虑在$\ell_2$距离下学习低度量子对象至$\varepsilon$误差的问题。我们展示了以下结果：$(i)$ 未知的$n$量子比特度-$d$（在泡利基下）量子通道和幺正算符可以使用$O(1/\varepsilon^d)$次查询（与$n$无关）进行学习；$(ii)$ 由$d$次查询量子算法产生的多项式$p:\{-1,1\}^n\rightarrow [-1,1]$可以从$O((1/\varepsilon)^d\cdot \log n)$个随机样本$(x,p(x))$中经典地学习（这意味着即使对于$d=O(\log n)$也具有可学习性）；$(iii)$ 度-$d$多项式$p:\{-1,1\}^n\to [-1,1]$可以通过对块编码$p$的量子幺正算符$U_p$进行$O(1/\varepsilon^d)$次查询来学习。我们的主要技术贡献是新的用于量子通道和完全有界多项式的Bohnenblust-Hille不等式。