We initiate the study of quantum agnostic learning of phase states with respect to a function class $\mathsf{C}\subseteq \{c:\{0,1\}^n\rightarrow \{0,1\}\}$: given copies of an unknown $n$-qubit state $|ψ\rangle$ which has fidelity $\textsf{opt}$ with a phase state $|φ_c\rangle=\frac{1}{\sqrt{2^n}}\sum_{x\in \{0,1\}^n}(-1)^{c(x)}|x\rangle$ for some $c\in \mathsf{C}$, output $|φ\rangle$ which has fidelity $|\langle φ| ψ\rangle|^2 \geq \textsf{opt}-\varepsilon$. To this end, we give agnostic learning protocols for the following classes: (i) Size-$t$ decision trees which runs in time $\textsf{poly}(n,t,1/\varepsilon)$. This also implies $k$-juntas can be agnostically learned in time $\textsf{poly}(n,2^k,1/\varepsilon)$. (ii) $s$-term DNF formulas in time $\textsf{poly}(n,(s/\varepsilon)^{\log \log (s/\varepsilon) \cdot \log(1/\varepsilon)})$. Our main technical contribution is a quantum agnostic boosting protocol which converts a weak agnostic learner, which outputs a parity state $|φ\rangle$ such that $|\langle φ|ψ\rangle|^2\geq \textsf{opt}/\textsf{poly}(n)$, into a strong learner which outputs a superposition of parity states $|φ'\rangle$ such that $|\langle φ'|ψ\rangle|^2\geq \textsf{opt} - \varepsilon$. Using quantum agnostic boosting, we obtain a $n^{O(\log(n/\varepsilon) \cdot \log \log n)}$-time algorithm for $\varepsilon$-learning $\textsf{poly}(n)$-sized depth-$3$ circuits (consisting of $\textsf{AND}$, $\textsf{OR}$, $\textsf{NOT}$ gates) in the uniform $\textsf{PAC}$ model given quantum examples. Classically, obtaining an algorithm with a similar complexity has been an open question in the $\textsf{PAC}$ model and our work answers this given quantum examples.

翻译：我们针对函数类 $\mathsf{C}\subseteq \{c:\{0,1\}^n\rightarrow \{0,1\}\}$ 启动了相位态的量子不可知学习研究：给定未知 $n$ 量子比特态 $|ψ\rangle$ 的若干副本，该态与某个 $c\in \mathsf{C}$ 对应的相位态 $|φ_c\rangle=\frac{1}{\sqrt{2^n}}\sum_{x\in \{0,1\}^n}(-1)^{c(x)}|x\rangle$ 具有保真度 $\textsf{opt}$，目标是输出一个态 $|φ\rangle$，使其满足 $|\langle φ| ψ\rangle|^2 \geq \textsf{opt}-\varepsilon$。为此，我们为以下类提供了不可知学习协议：(i) 规模为 $t$ 的决策树，其运行时间为 $\textsf{poly}(n,t,1/\varepsilon)$。这也意味着 $k$-juntas 可以在 $\textsf{poly}(n,2^k,1/\varepsilon)$ 时间内被不可知学习。(ii) $s$ 项 DNF 公式，其运行时间为 $\textsf{poly}(n,(s/\varepsilon)^{\log \log (s/\varepsilon) \cdot \log(1/\varepsilon)})$。我们的主要技术贡献是一个量子不可知增强协议，该协议将弱不可知学习器（输出一个满足 $|\langle φ|ψ\rangle|^2\geq \textsf{opt}/\textsf{poly}(n)$ 的奇偶态 $|φ\rangle$）转换为强学习器，后者输出一个奇偶态的叠加 $|φ'\rangle$，使得 $|\langle φ'|ψ\rangle|^2\geq \textsf{opt} - \varepsilon$。利用量子不可知增强，我们获得了一个在均匀 $\textsf{PAC}$ 模型中给定量子样本时，以 $n^{O(\log(n/\varepsilon) \cdot \log \log n)}$ 时间 $\varepsilon$-学习规模为 $\textsf{poly}(n)$ 的深度-$3$ 电路（由 $\textsf{AND}$、$\textsf{OR}$、$\textsf{NOT}$ 门组成）的算法。在经典情况下，在 $\textsf{PAC}$ 模型中获取具有类似复杂度的算法一直是一个开放性问题，而我们的工作在给定量子样本的条件下回答了该问题。