[论文解读] Sample Efficient Algorithms for Learning Quantum Channels in PAC Model and the Approximate State Discrimination Problem

Kai-Min Chung, Han-Hsuan Lin|arXiv (Cornell University)|Oct 25, 2018

Machine Learning and Algorithms参考文献 33被引用 9

一句话总结

本文提出了一种用于量子通道的PAC学习框架，引入了两种样本高效的算法：一种用于纯态输出，样本复杂度为 $ O((\log |C| + \log(1/\delta))/\epsilon^2) $；另一种用于混合态输出，复杂度为 $ O((\log^3 |C|)(\log |C| + \log(1/\delta))/\epsilon^2) $。关键贡献在于通过近似态判别实现高效量子过程学习，在结构化设置下相比朴素量子过程层析实现了指数级改进。

ABSTRACT

The probably approximately correct (PAC) model [Leslie G. Valiant, 1984] is a well studied model in classical learning theory. Here, we generalize the PAC model from concepts of Boolean functions to quantum channels, introducing PAC model for learning quantum channels, and give two sample efficient algorithms that are analogous to the classical "Occam’s razor" result [Blumer et al., 1987]. The classical Occam’s razor algorithm is done trivially by excluding any concepts not compatible with the input-output pairs one gets, but such an approach is not immediately possible with a concept class of quantum channels, because the outputs are unknown quantum states from the quantum channel. To study the quantum state learning problem associated with PAC learning quantum channels, we focus on the special case where the channels all have constant output. In this special case, learning the channels reduce to a problem of learning quantum states that is similar to the well known quantum state discrimination problem [Joonwoo Bae and Leong-Chuan Kwek, 2017], but with the extra twist that we allow ε-trace-distance-error in the output. We call this problem Approximate State Discrimination, which we believe is a natural problem that is of independent interest. We give two algorithms for learning quantum channels in PAC model. The first algorithm has sample complexity O((log|C| + log(1/ δ))/(ε²)), but only works when the outputs are pure states, where C is the concept class, ε is the error of the output, and δ is the probability of failure of the algorithm. The second algorithm has sample complexity O((log³|C|(log|C|+log(1/ δ)))/(ε²)), and work for mixed state outputs. Some implications of our results are that we can PAC-learn a polynomial sized quantum circuit in polynomial samples, and approximate state discrimination can be solved in polynomial samples even when the size of the input set is exponential in the number of qubits, exponentially better than a naive state tomography.

研究动机与目标

• 将经典的PAC学习模型推广至量子通道，实现对未知量子过程的高效学习。
• 解决当输出态为未知量子系统时学习量子通道的挑战，利用量子态判别技术。
• 提出并求解近似态判别问题，该问题为允许迹距离误差的量子态判别变体。
• 建立纯态输出与混合态输出量子通道学习的样本复杂度边界，且不依赖于输出维数。
• 在结构化设置下，展示相比标准量子过程层析的指数级样本节省。

提出的方法

• 通过迭代剔除与采样输入-输出对不一致的假设，将经典的奥卡姆剃刀原理推广至量子通道。
• 利用量子态判别技术，测试候选通道与观测输出态的兼容性，即使输出为未知量子态亦可。
• 采用迹距离作为量子通道之间的距离度量，对输入分布 $ D $ 取平均。
• 应用切尔诺夫不等式与联合界控制假设选择中的误差概率。
• 在混合态情况下，使用递归或迭代测量策略，从多个输出态副本中提取区分信息。
• 利用 $ \epsilon $-打包网与霍尔沃夫信息界推导样本复杂度的信息论下界。

实验结果

研究问题

• RQ1 PAC学习框架能否从经典概念推广至量子通道？
• RQ2 当输出态为未知量子系统时，如何高效学习量子通道？
• RQ3 学习具有纯态输出的量子通道的最优样本复杂度是多少？
• RQ4 学习具有混合态输出的量子通道的最优样本复杂度是多少？
• RQ5 近似态判别能否被高效求解，其与量子过程学习的关系如何？

主要发现

• 首个算法在纯态输出下实现样本复杂度 $ O((\log |C| + \log(1/\delta))/\epsilon^2) $，在对数因子范围内与经典奥卡姆剃刀界匹配。
• 第二个算法在混合态输出下实现样本复杂度 $ O((\log^3 |C|)(\log |C| + \log(1/\delta))/\epsilon^2) $，且不依赖于输出维数。
• 结果表明，任何大小为多项式级的 $ n $ qubit 量子电路均可在 $ \text{poly}(n) $ 个样本内被PAC学习，相比朴素过程层析实现指数级改进。
• 近似态判别问题即使在输入集大小随量子比特数指数增长时，也可在多项式样本内求解。
• 建立了纯态近似判别问题的下界 $ \Omega((1 - \delta)\ln |C| / \epsilon^2) / \ln(\ln |C| / \epsilon) $，与上界在对数因子范围内一致。
• 证明了对 $ d $ 维空间上经典分布的广义学习样本复杂度下界为 $ \Omega(\sqrt{d}) $，表明在高维经典分布中高效广义学习存在局限。