精确学习的最优量子-经典分离

中文摘要 AI 辅助

我们研究了概念类 $\mathcal C\subseteq\{0,1\}^N$ 的带成员查询的精确学习，重点关注其确定性、随机化和量子查询复杂度之间的关系，分别记为 $\mathsf{D}(\mathcal C)$、$\mathsf{R}(\mathcal C)$ 和 $\mathsf{Q}(\mathcal C)$。该模型中两个典型的量子加速分别由 Grover 搜索和 Bernstein-Vazirani 算法体现，这导致了长期存在的猜想：$$ \mathsf{R}(\mathcal C)=O(\mathsf{Q}(\mathcal C)^2+\mathsf{Q}(\mathcal C)\log N)。 $$ 我们首先通过构造概念类 $\mathcal C$ 和 $\mathcal C'$ 反驳了这一猜想，它们满足 \\[ \mathsf{R}(\mathcal C)=\Omega\\!\left(\frac{\mathsf{Q}(\mathcal C)^3\log N}{\log \mathsf{Q}(\mathcal C)}\right) \qquad\text{且}\qquad \mathsf{D}(\mathcal C')=\Omega(\mathsf{Q}(\mathcal C')^3\log N)。 \\] 第一个界与 Arunachalam 等人 [Quantum'21] 的上界在常数因子内匹配，而第二个界与 Servedio 和 Gortler [SICOMP'04] 的上界匹配。特别是，这表明 Arunachalam 等人的随机化上界中的节省从根本上依赖于随机性。除了刻画经典与量子查询复杂度之间的最优关系外，我们的结果首次表明学习中的量子加速可以超越 Grover 和 Bernstein-Vazirani 范式。

英文摘要

We study exact learning with membership queries for concept classes $\mathcal C\subseteq\{0,1\}^N$, focusing on the relationships among their deterministic, randomized, and quantum query complexities, denoted $\mathsf{D}(\mathcal C)$, $\mathsf{R}(\mathcal C)$, and $\mathsf{Q}(\mathcal C)$, respectively. The two canonical quantum speedups in this model are witnessed by Grover search and Bernstein-Vazirani, leading to the longstanding conjecture $$ \mathsf{R}(\mathcal C)=O(\mathsf{Q}(\mathcal C)^2+\mathsf{Q}(\mathcal C)\log N). $$ We first refute this conjecture by constructing concept classes $\mathcal C$ and $\mathcal C'$ satisfying \[ \mathsf{R}(\mathcal C)=Ω\!\left(\frac{\mathsf{Q}(\mathcal C)^3\log N}{\log \mathsf{Q}(\mathcal C)}\right) \qquad\text{and}\qquad \mathsf{D}(\mathcal C')=Ω(\mathsf{Q}(\mathcal C')^3\log N). \] The first bound matches the upper bound of Arunachalam et al.~[Quantum'21] up to constant factors, while the second matches the upper bound of Servedio and Gortler~[SICOMP'04]. In particular, this shows that the saving in the randomized upper bound of Arunachalam et al. fundamentally relies on randomness. Apart from characterizing the optimal relationship between classical and quantum query complexity, our results are the first to show that quantum speedups for learning can go beyond the Grover and Bernstein-Vazirani paradigms.

发表机构

• IBM Research（IBM研究院）
• University of Maryland（马里兰大学）
• University of Liverpool（利物浦大学）

机构由 AI 辅助整理，请以论文原文为准。