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.

翻译：我们研究了在实际测量约束下高精度区域中影子层析成像的样本复杂度。给定一个未知的 $d$ 维量子态 $ρ$ 和一组已知的观测量 $\{O_i\}_{i=1}^m$，目标是以 $L_p$ 范数精度 $ε$ 估计期望值 $\{\mathrm{tr}(O_iρ)\}_{i=1}^m$，使用的测量可以是自适应的，但每次仅作用于 $O(\mathrm{polylog}(d))$ 个 $ρ$ 的副本。我们关注 $ε$ 低于一个实例相关阈值的区域。我们的主要贡献是给出了样本复杂度的实例最优刻画，即 $\tildeΘ(Γ_p/ε^2)$，其中 $Γ_p$ 是通过一个涉及逆费希尔信息矩阵的优化公式定义的 $\{O_i\}_{i=1}^m$ 的函数。此前，仅在特殊情况下已知紧界，例如具有 $L_\infty$ 范数误差的泡利影子层析成像。具体而言，我们首先分析一个更简单的非自适应变体，其目标是在测量完成后，估计一个形式为 $\sum_{i=1}^m α_i O_i$（其中 $\|α\|_q = 1$，$q$ 是 $p$ 的对偶范数）的观测量。对于单拷贝测量，我们获得了 $Θ(Γ^{\mathrm{ob}}_p/ε^2)$ 的样本复杂度。然后我们证明 $\tildeΘ(Γ_p/ε^2)$ 对于原始问题是必要且充分的，其下界适用于无偏、有界的估计器。我们的上界依赖于一个结合粗粒度层析成像与局部估计的两步算法。值得注意的是，$Γ^{\mathrm{ob}}_\infty = Γ_\infty$。在两种情况下，允许 $c$ 拷贝测量最多能将样本复杂度改进 $Ω(1/c)$。我们的结果建立了量子学习与计量学之间的定量对应关系，将渐近计量极限与有限样本学习保证统一起来。