Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach

We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown $d$-dimensional quantum state $\rho$ and a known set of observables $\{O_i\}_{i=1}^m$, the goal is to estimate expectation values $\{\mathrm{tr}(O_i\rho)\}_{i=1}^m$ to accuracy $\epsilon$ in $L_p$-norm, using possibly adaptive measurements that act on $O(\mathrm{polylog}(d))$ number of copies of $\rho$ at a time. We focus on the regime where $\epsilon$ is below an instance-dependent threshold. Our main contribution is an instance-optimal characterization of the sample complexity as $\tilde{\Theta}(\Gamma_p/\epsilon^2)$, where $\Gamma_p$ 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 $\sum_{i=1}^m \alpha_i O_i$ with $\|\alpha\|_q = 1$ (where $q$ is dual to $p$) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of $\Theta(\Gamma^{\mathrm{ob}}_p/\epsilon^2)$. We then show $\tilde{\Theta}(\Gamma_p/\epsilon^2)$ 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, $\Gamma^{\mathrm{ob}}_\infty = \Gamma_\infty$. In both cases, allowing $c$-copy measurements improves the sample complexity by at most $\Omega(1/c)$. Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.