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

2026-02-04

AI Summary

This study addresses the problem of efficiently estimating the expectation values of a set of known observables on an unknown quantum state under the constraints of high accuracy and limited copies per measurement. To this end, the authors propose a two-stage adaptive algorithm that combines coarse-grained tomography with local estimation, tailored for scenarios where each measurement acts on only a small number of state copies. They establish the first instance-optimal characterization of sample complexity: the total complexity scales as ~Θ(Γₚ/ε²), where Γₚ depends on the observable set, and in the single-copy setting it is Θ(Γₚ^ob/ε²). Furthermore, they prove that c-copy measurements yield at most an Ω(1/c) advantage in sample complexity. This work unifies asymptotic quantum metrology limits with finite-sample learning guarantees, establishing a quantitative bridge between quantum learning and quantum metrology.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

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.

Problem

• shadow tomography
• sample complexity
• quantum metrology
• high-precision estimation
• few-copy measurements

Innovation

• shadow tomography
• instance-optimal
• Fisher information
• quantum metrology
• sample complexity