发表机构
The University of Tokyo（东京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明在量子PAC学习中，仅前向访问态制备酉算子无法在查询复杂度上优于经典数据或量子数据副本，逆访问是实现优势的关键。
AI 中文摘要
量子计算能否减少从未知概率分布中采样以学习预测规则所需的数据量，是量子机器学习中的一个基本问题。量子PAC学习利用量子数据（即量子态，其振幅平方编码了经典学习数据所采样的未知分布）来研究这一问题。仅凭此类量子数据的副本，最优最坏情况样本复杂度在渐近意义上与经典PAC学习相匹配。相比之下，同时访问该态的态制备酉算子及其逆，可以在可实现学习中改善查询复杂度对精度参数的依赖。然而，仅前向访问是否允许这种改善仍不清楚。在本工作中，考虑兼容态制备酉算子及其有限环境维度的最坏情况，我们证明可实现学习和不可知学习的最优仅前向查询复杂度分别为$\Theta((d+\log(1/\delta))/\varepsilon)$和$\Theta((d+\log(1/\delta))/\varepsilon^2)$，其中$d$是概念类的VC维，$\varepsilon$是精度参数，$\delta$是失败概率。这些界限与使用经典数据或量子数据副本的最优样本复杂度相匹配。为证明这些结果，我们建立了一个归约，利用制备态的$q$个副本近似任何$q$查询仅前向算法的Haar平均输出。这些结果表明，在这种最坏情况设置下，仅前向访问无法提供相对于从经典数据或量子数据副本学习的渐近查询复杂度优势，并确立了逆访问在已知可实现设置改善中的关键作用。我们的归约还为通过态副本下界分析前向态制备访问的限制提供了新框架。
英文摘要
Whether quantum computation can reduce the amount of data sampled from an unknown probability distribution required to learn a prediction rule is a fundamental question in quantum machine learning. Quantum PAC learning studies this question using quantum data as a quantum state whose squared amplitudes encode the unknown distribution from which classical learning data are sampled. With only copies of such quantum data, the optimal worst-case sample complexity asymptotically matches that of classical PAC learning. In contrast, access to both a state-preparation unitary for this state and its inverse can improve the query-complexity dependence on the accuracy parameter in realizable learning. However, it has remained unclear whether forward-only access allows such an improvement. In this work, taking the worst case over compatible state-preparation unitaries and their finite ambient dimensions, we show that the optimal forward-only query complexities of realizable and agnostic learning are, respectively, $Θ((d+\log(1/δ))/\varepsilon)$ and $Θ((d+\log(1/δ))/\varepsilon^2)$, where $d$ is the VC dimension of the concept class, $\varepsilon$ the accuracy parameter, and $δ$ the failure probability. These bounds match the optimal sample complexities with classical data or quantum data copies. To prove them, we establish a reduction using $q$ copies of the prepared state to approximate the Haar-averaged output of any $q$-query forward-only algorithm. These results show that forward-only access cannot provide an asymptotic query-complexity advantage over learning from classical data or quantum data copies in this worst-case setting, and establish the essential role of inverse access in the known realizable-setting improvement. Our reduction also provides a new framework for analyzing limitations of forward state-preparation access via state-copy lower bounds.
Comments40 pages, no figures
