提出高精度量子态层析的最优采样方法，实现低误差测量。
Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach
- 基于计量学框架，设计自适应测量策略优化样本使用。
- 理论证明样本复杂度为 Γₚ/ε² 阶，优于以往特定情况结果。
- 适用于需要高精度估计的量子实验与算法开发人员。
研究在现实测量约束下，高精度量子态层析的样本复杂度。给定未知的 d 维量子态 ρ 和一组已知可观测量 {O_i}，目标是在最多使用 O(polylog(d)) 个副本的自适应测量条件下，以 L_p-范数精度 ε 估计所有期望值 {tr(O_iρ)}。关注 ε 低于依赖实例的阈值的情形。主要贡献是首次获得样本复杂度的实例最优表征：˜Θ(Γ_p/ε²)，其中 Γ_p 由逆费舍尔信息矩阵定义的优化公式决定。此前仅在特殊情形（如 Pauli 层析和 L_∞ 范数）有紧界。我们先分析更简单的无偏情形：估计形如 ∑α_i O_i 且 ‖α‖_q=1（q 是 p 的对偶）的可观测量，单副本测量下复杂度为 Θ(Γ^ob_p/ε²)。随后证明原问题所需复杂度为 ˜Θ(Γ_p/ε²)，下界适用于无偏、有界估计器。上界依赖两步算法：粗略层析结合局部估计。值得注意的是 Γ^ob_∞ = Γ_∞。允许 c 副本测量最多将复杂度降低 Ω(1/c)。结果建立了量子学习与计量学间的定量对应，统一了渐近计量极限与有限样本学习保证。
原文摘要 · Abstract (English)
We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown $d$-dimensional quantum state $ρ$ and a known set of observables $\{O_i\}_{i=1}^m$, the goal is to estimate expectation values $\{\mathrm{tr}(O_iρ)\}_{i=1}^m$ to accuracy $ε$ in $L_p$-norm, using possibly adaptive measurements that act on $O(\mathrm{polylog}(d))$ number of copies of $ρ$ at a time. We focus on the regime where $ε$ is below an instance-dependent threshold. Our main contribution is an instance-optimal characterization of the sample complexity as $\tildeΘ(Γ_p/ε^2)$, where $Γ_p$ is a function of $\{O_i\}_{i=1}^m$ defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with $L_\infty$-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form $\sum_{i=1}^m α_i O_i$ with $\|α\|_q = 1$ (where $q$ is dual to $p$) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of $Θ(Γ^{\mathrm{ob}}_p/ε^2)$. We then show $\tildeΘ(Γ_p/ε^2)$ is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably, $Γ^{\mathrm{ob}}_\infty = Γ_\infty$. In both cases, allowing $c$-copy measurements improves the sample complexity by at most $Ω(1/c)$. Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.
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