稳定态克隆与学习所需样本量同阶,均需Θ(n)。
Cloning is as Hard as Learning for Stabilizer States
- 用阿贝尔隐子群框架和随机纯化通道分析稳定态克隆。
- 证明n量子比特稳定态克隆最优样本复杂度为Θ(n)。
- 揭示量子学习与量子加密中的基础限制关系。
非正交量子态的不可克隆性是量子理论的基础。即使允许近似误差,克隆任意未知纯态所需的初始副本数也与完全学习该态所需数量相当。现代量子学习理论常考虑结构化态类,并利用其结构设计优于通用态层析的学习算法。这引出一个问题:对于此类结构化态,学习与克隆的样本复杂度有何关系?本文针对一个重要态类——n量子比特稳定态——给出答案:克隆的最优样本复杂度为Θ(n)。因此,对该类态而言,克隆同样等价于学习。为此,我们采用近期提出的阿贝尔隐子群框架中的表示论工具,结合新提出的结构化随机纯化通道,将稳定态克隆问题转化为经典学习理论中一类具有线性结构的概率分布样本放大问题。由此通过建立新的样本放大下界,导出克隆下界。本结果为无克隆定理提供了更精细视角,连接了量子基础、量子学习理论与量子密码学。
原文摘要 · Abstract (English)
The impossibility of simultaneously cloning non-orthogonal states lies at the foundations of quantum theory. Even when allowing for approximation errors, cloning an arbitrary unknown pure state requires as many initial copies as needed to fully learn the state. Rather than arbitrary unknown states, modern quantum learning theory often considers structured classes of states and exploits such structure to develop learning algorithms that outperform general-state tomography. This raises the question: How do the sample complexities of learning and cloning relate for such structured classes? We answer this question for an important class of states. Namely, for $n$-qubit stabilizer states, we show that the optimal sample complexity of cloning is $Θ(n)$. Thus, also for this structured class of states, cloning is as hard as learning. To prove these results, we use representation-theoretic tools in the recently proposed Abelian State Hidden Subgroup framework and a new structured version of the recently introduced random purification channel to relate stabilizer state cloning to a variant of the sample amplification problem for probability distributions that was recently introduced in classical learning theory. This allows us to obtain our cloning lower bounds by proving new sample amplification lower bounds for classes of distributions with an underlying linear structure. Our results provide a more fine-grained perspective on No-Cloning theorems, opening up connections from foundations to quantum learning theory and quantum cryptography.
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