In this work, we consider the problems of learning junta distributions, their quantum counterparts (quantum junta states) and $\mathsf{QAC}^0$ circuits, which we show to be close to juntas. (1) Junta distributions. A probability distribution $p:\{-1,1\}^n\to \mathbb [0,1]$ is a $k$-junta if it only depends on $k$ bits. We show that they can be learned with to error $\varepsilon$ in total variation distance from $O(2^k\log(n)/\varepsilon^2)$ samples, which quadratically improves the upper bound of Aliakbarpour et al. (COLT'16) and matches their lower bound in every parameter. (2) Junta states. We initiate the study of $n$-qubit states that are $k$-juntas, those that are the tensor product of a $k$-qubit state and an $(n-k)$-qubit maximally mixed state. We show that these states can be learned with error $\varepsilon$ in trace distance with $O(12^{k}\log(n)/\varepsilon^2)$ single copies. We also prove a lower bound of $Ω((4^k+\log (n))/\varepsilon^2)$ copies. Additionally, we show that, for constant $k$, $\tildeΘ(2^n/\varepsilon^2)$ copies are necessary and sufficient to test whether a state is $\varepsilon$-close or $7\varepsilon$-far from being a $k$-junta. (3) $\mathsf{QAC}^0$ circuits. Nadimpalli et al. (STOC'24) recently showed that the Pauli spectrum of $\mathsf{QAC}^0$ circuits (with a limited number of auxiliary qubits) is concentrated on low-degree. We remark that they implied something stronger, namely that the Choi states of those circuits are close to be juntas. As a consequence, we show that $n$-qubit $\mathsf{QAC}^0$ circuits with size $s$, depth $d$ and $a$ auxiliary qubits can be learned from $2^{O(\log(s^22^a)^d)}\log (n)$ copies of the Choi state, improving the $n^{O(\log(s^22^a)^d)}$ by Nadimpalli et al.

翻译：本文研究junta分布、其量子对应物（量子junta态）以及与junta接近的$\mathsf{QAC}^0$电路的学习问题。（1）Junta分布。概率分布$p:\{-1,1\}^n\to \mathbb [0,1]$若仅依赖于$k$个比特，则称为$k$-junta分布。我们证明可通过$O(2^k\log(n)/\varepsilon^2)$个样本以全变差距离误差$\varepsilon$学习该类分布，该结果在二次改进Aliakbarpour等人（COLT'16）上界的同时与所有参数下的下界匹配。（2）Junta态。我们首次研究$k$-junta的$n$量子比特态，即$k$量子比特态与$(n-k)$量子比特最大混合态的张量积。我们证明可通过$O(12^{k}\log(n)/\varepsilon^2)$个单副本以迹距离误差$\varepsilon$学习这些态，同时证明下界为$\Omega((4^k+\log (n))/\varepsilon^2)$个副本。此外，对于常数$k$，我们证明需$\tildeΘ(2^n/\varepsilon^2)$个副本才能以$\varepsilon$-接近或$7\varepsilon$-远离的条件判定某态是否为$k$-junta。（3）$\mathsf{QAC}^0$电路。Nadimpalli等人（STOC'24）近期证明了$\mathsf{QAC}^0$电路（具有有限辅助量子比特）的泡利谱集中于低阶。我们指出他们隐含了更强结论：这些电路的Choi态接近junta态。据此，我们证明对于规模为$s$、深度为$d$且含$a$个辅助量子比特的$n$量子比特$\mathsf{QAC}^0$电路，可通过$2^{O(\log(s^22^a)^d)}\log (n)$个Choi态副本学习，改进了Nadimpalli等人的$n^{O(\log(s^22^a)^d)}$结果。