arXiv 1802.090252018-02quant-phcs.LG

量子态的在线学习

Online Learning of Quantum States

UT Austin（得克萨斯大学奥斯汀分校）；Princeton University（普林斯顿大学）；Google AI Princeton（谷歌人工智能普林斯顿分部）；Google AI, New York（谷歌人工智能纽约分部）；University of Waterloo（滑铁卢大学）

Scott Aaronson, Xinyi Chen, Elad Hazan, Satyen Kale, Ashwin Nayak

备注：: 18 pages
DOI：: 10.1088/1742-5468/ab3988

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摘要：

假设我们有一个未知的$n$ -量子位态$ρ$的许多副本。我们使用已知的双结果测量$E_{1}$测量$ρ$的一些拷贝，然后使用测量$E_{2}$测量其他拷贝，等等。在每个阶段$t$，我们使用先前测量的结果生成关于状态$ρ$的当前假设$σ_{t}$。我们表明，这是可能做到这一点的方式，保证$|\operatorname{Tr}(E_{i} σ_{t}) - \operatorname{Tr}(E_{i}ρ) |$，在我们的预测下一个测量的错误，是至少$\varepsilon$在最多$\operatorname{O}\!\left(n / \varepsilon^2 \right) $倍。即使在“不可实现”的设置-在测量结果中可能存在任意噪声-我们展示了如何输出假设状态，这些假设状态在第一次$T$测量时最多$\operatorname{O}\!\left(\sqrt {Tn}\right) $次明显比最佳可能状态差。这些结果将Aaronson在2007年提出的关于量子态的PAC可学习性的定理推广到在线和遗憾最小化设置。我们给出了三种不同的方法来证明我们的结果-使用凸优化，量子后选择，和顺序脂肪粉碎尺寸-这在参数和可移植性方面有不同的优势。

英文摘要：

Suppose we have many copies of an unknown $n$-qubit state $ρ$. We measure some copies of $ρ$ using a known two-outcome measurement $E_{1}$, then other copies using a measurement $E_{2}$, and so on. At each stage $t$, we generate a current hypothesis $σ_{t}$ about the state $ρ$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\operatorname{Tr}(E_{i} σ_{t}) - \operatorname{Tr}(E_{i}ρ) |$, the error in our prediction for the next measurement, is at least $\varepsilon$ at most $\operatorname{O}\!\left(n / \varepsilon^2 \right) $ times. Even in the "non-realizable" setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that do significantly worse than the best possible states at most $\operatorname{O}\!\left(\sqrt {Tn}\right) $ times on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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