Suppose we have many copies of an unknown $n$-qubit state $\rho$. We measure
some copies of $\rho$ 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 $\sigma_{t}$ about the state $\rho$, using the outcomes of
the previous measurements. We show that it is possible to do this in a way that
guarantees that $|\operatorname{Tr}(E_{i} \sigma_{t}) -
\operatorname{Tr}(E_{i}\rho) |$, 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.

该研究论文研究了如何通过量子状态的测量来生成假设，以指导下一次测量的选取，即减少答案预测失误率。

量子态的在线学习

Online Learning of Quantum States

Using data obtained in a controlled ad-auction experiment that we ran, we
evaluate the regret-based approach to econometrics that was recently suggested
by Nekipelov, Syrgkanis, and Tardos (EC 2015). We found that despite the weak
regret-based assumptions, the results were (at least) as accurate as those
obtained using classic equilibrium-based assumptions. En route we studied to
what extent humans actually minimize regret in our ad auction, and found a
significant difference between the "high types" (players with a high valuation)
who indeed rationally minimized regret and the "low types" who significantly
overbid. We suggest that correcting for these biases and adjusting the
regret-based econometric method may improve the accuracy of estimated values.

该研究使用实验数据对 Nekipelov、Syrgkanis 和 Tardos（EC 2015）最近提出的基于遗憾的计量经济学方法进行了评估，发现尽管采用了较弱的遗憾假设，其结果至少与采用经典均衡假设获得的结果一样准确。其中，研究发现参与广告拍卖的高估价者理性地将遗憾降至最低，而低估价者的竞价则存在显著超额现象，作者建议进行修正这些偏差并调整基于遗憾的计量经济方法，以提高预测的准确性。