Symmetry underlies many of the most effective classical and quantum learning algorithms, yet whether quantum learners can gain a fundamental advantage under symmetry-imposed structures remains an open question. Based on evidence that classical statistical query ($\SQ$) frameworks have revealed exponential query complexity in learning symmetric function classes, we ask: can quantum learning algorithms exploit the problem symmetry better? In this work, we investigate the potential benefits of symmetry within the quantum statistical query ($\QSQ$) model, which is a natural quantum analog of classical $\SQ$. Our results uncover three distinct phenomena: (i) we obtain an exponential separation between $\QSQ$ and $\SQ$ on a permutation-invariant function class; (ii) we establish query complexity lower bounds for $\QSQ$ learning that match, up to constant factors, the corresponding classical $\SQ$ lower bounds for most commonly studied symmetries; however, the potential advantages may occur under highly skewed orbit distributions; and (iii) we further identify a tolerance-based separation exists, where quantum learners succeed at noise levels that render classical $\SQ$ algorithms ineffective. Together, these findings provide insight into when symmetry can enable quantum advantage in learning.
Symmetries and optical transitions of hexagonal quantum dots in GaAs/AlGaAs nanowires
我们研究了AlGaAs纳米线内六角GaAs量子点的电子态和光跃迁性质。这种点由于其高度对称性而特别有趣。开发了一种基于类算子（PTCO）的简化后对称化技术，该技术使人们能够在一次运行中从Dalessi等人[Phys. Rev. B 81，125106（2010）]在求解薛定谔方程后报告的最大对称化和场约简（MSRF）方法带来的洞察力中获益。PTCO的明确优势是，它不需要修改任何现有的代码来计算电子结构，并且它允许对提升的对称性进行数值测试。我们在4带k.p模型的框架中表明，尽管纳米结构的D6h对称性在微观水平上被底层的闪锌矿晶体结构破坏，但效果相当小。电子态及其光发射的大多数特殊性可以通过对称性提升到D6 h以及近似方位角和径向量子数的存在来理解。
Simons Institute for the Theory of Computing, University of California at Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；Institute for Theoretical Physics, ETH Zürich（苏黎世联邦理工学院理论物理研究所）
