研究量子实验中自适应选择观测值的预测难题，揭示了采样数量的理论下界。
Predicting adaptively chosen observables in quantum systems
- 针对自适应选择的局域与泡利观测值，证明需至少√M次采样
- 提出高效算法实现√M采样下界，避免错误预测
- 对有界弗罗贝尼乌斯范数观测值，仅需log M次采样
最近的研究表明，仅需$\mathcal{O}(\log M)$次测量即可预测任意大量子多体系统的$M$个性质。然而，这些结果假设待预测性质是独立于数据选定的。在实际中，科学家可能根据先前预测结果自适应地选择新性质，这一假设被破坏。本文研究了三类可观测量（局域、泡利、有界弗罗贝尼乌斯范数）在自适应设置下的预测问题。我们证明，对于$M$个自适应选择的局域和泡利可观测量，$Ω(\sqrt{M})$次样本是预测其期望值所必需的。同时，我们设计出可计算高效的算法，达到这一信息论下界。相比之下，对于有界弗罗贝尼乌斯范数可观测量，我们提出一种算法仅需$\mathcal{O}(\log M)$次采样，且不依赖系统大小。结果揭示了自适应性在量子实验数据分析中的潜在风险，并提供了新的算法工具以防范错误预测。
原文摘要 · Abstract (English)
Recent advances have demonstrated that $\mathcal{O}(\log M)$ measurements suffice to predict $M$ properties of arbitrarily large quantum many-body systems. However, these remarkable findings assume that the properties to be predicted are chosen independently of the data. This assumption can be violated in practice, where scientists adaptively select properties after looking at previous predictions. This work investigates the adaptive setting for three classes of observables: local, Pauli, and bounded-Frobenius-norm observables. We prove that $Ω(\sqrt{M})$ samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of $M$ adaptively chosen local and Pauli observables. We also present computationally-efficient algorithms that achieve this information-theoretic lower bound. In contrast, for bounded-Frobenius-norm observables, we devise an algorithm requiring only $\mathcal{O}(\log M)$ samples, independent of system size. Our results highlight the potential pitfalls of adaptivity in analyzing data from quantum experiments and provide new algorithmic tools to safeguard against erroneous predictions in quantum experiments.
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