[论文解读] Quantum statistical query learning

Srinivasan Arunachalam, Alex B. Grilo+1 more|arXiv (Cornell University)|Feb 19, 2020

Machine Learning and Algorithms参考文献 56被引用 19

一句话总结

本文提出了量子统计查询（QSQ）学习模型，这是一种受限但强大的框架，其中经典学习者仅通过统计查询访问量子样本——即对量子可观测量的期望值进行估计。关键贡献在于，QSQ学习能高效处理经典SQ模型下难以学习的概念类，如奇偶函数、O(log n)-juntas以及多项式大小的DNF公式，展示了即使在此类受限设置下仍存在量子优势。

ABSTRACT

We propose a learning model called the quantum statistical learning QSQ model, which extends the SQ learning model introduced by Kearns to the quantum setting. Our model can be also seen as a restriction of the quantum PAC learning model: here, the learner does not have direct access to quantum examples, but can only obtain estimates of measurement statistics on them. Theoretically, this model provides a simple yet expressive setting to explore the power of quantum examples in machine learning. From a practical perspective, since simpler operations are required, learning algorithms in the QSQ model are more feasible for implementation on near-term quantum devices. We prove a number of results about the QSQ learning model. We first show that parity functions, (log n)-juntas and polynomial-sized DNF formulas are efficiently learnable in the QSQ model, in contrast to the classical setting where these problems are provably hard. This implies that many of the advantages of quantum PAC learning can be realized even in the more restricted quantum SQ learning model. It is well-known that weak statistical query dimension, denoted by WSQDIM(C), characterizes the complexity of learning a concept class C in the classical SQ model. We show that log(WSQDIM(C)) is a lower bound on the complexity of QSQ learning, and furthermore it is tight for certain concept classes C. Additionally, we show that this quantity provides strong lower bounds for the small-bias quantum communication model under product distributions. Finally, we introduce the notion of private quantum PAC learning, in which a quantum PAC learner is required to be differentially private. We show that learnability in the QSQ model implies learnability in the quantum private PAC model. Additionally, we show that in the private PAC learning setting, the classical and quantum sample complexities are equal, up to constant factors.

研究动机与目标

• 形式化一种量子学习模型，通过统计查询限制对量子样本的访问，从而在近期量子设备上实现可行性。
• 探究当无法直接访问量子态，仅能获取其统计属性时，量子样本是否仍能提供学习优势。
• 建立QSQ学习、量子私有PAC学习与通信复杂性之间的联系。
• 利用信息理论度量（如弱统计查询维数）刻画QSQ模型中概念类的可学习性。
• 探究纠缠或叠加查询是否为实现量子学习优势所必需。

提出的方法

• 将QSQ模型提出为量子PAC模型的限制形式，学习者仅接收对可观测量在量子样本上期望值的估计。
• 使用形式为 |ψc*⟩ = Σx √D(x) |x, c*(x)⟩ 的量子样本，并查询赫米特可观测量M的期望值 ⟨ψc*|M|ψc*⟩。
• 应用量子查询复杂度与统计查询学习的技术分析可学习性，包括采用幅值放大与集中不等式。
• 引入概率表示维数（PRDIM）概念，以界定QSQ学习中的样本复杂度。
• 通过基于 log(WeakSQDIM(C)) 的下界，建立QSQ学习与小偏差量子通信模型之间的联系。
• 证明QSQ可学习性蕴含在私有量子PAC模型中的可学习性，并表明在私有设置下，经典与量子样本复杂度在常数因子范围内等价。

实验结果

研究问题

• RQ1 当学习者无法访问完整量子态，仅能获取其统计属性时，量子样本是否仍能提供学习优势？
• RQ2 奇偶函数与DNF公式等概念类是否可在QSQ模型中被高效学习，尽管它们在经典SQ模型中是困难的？
• RQ3 弱统计查询维数的对数是否为某些概念类的QSQ学习复杂度提供紧致下界？
• RQ4 QSQ学习能否用于实现差分隐私的量子PAC学习，其样本复杂度与经典私有学习相比如何？
• RQ5 在均匀分布下，是否存在量子PAC学习与QSQ学习之间的分离，暗示可能需要纠缠或叠加查询才能实现完整量子优势？

主要发现

• 奇偶函数、O(log n)-juntas以及多项式大小的DNF公式在QSQ模型中可被高效学习，而它们在经典SQ模型中是困难的。
• log(WeakSQDIM(C)) 提供了QSQ学习复杂度的下界，且对某些概念类而言该下界是紧致的。
• 该下界也对乘积分布下的小偏差量子通信模型产生了强有力的下界。
• QSQ可学习性蕴含在量子私有PAC模型中的可学习性，且在私有设置下，经典与量子样本复杂度在常数因子范围内等价。
• 本文未解决在均匀分布下量子PAC学习与QSQ学习之间是否存在分离，提示实现完整量子优势可能需要纠缠或叠加查询。
• 该框架为量子机器学习算法提供统一视角，因为许多已知的量子算法均可重新表述为统计查询模型的形式。