Optimal Shadow Estimation with Minimal Measurement Settings

• URL: http://arxiv.org/abs/2606.20003v1
• Date: Thu, 18 Jun 2026 09:40:17 GMT
• Title: Optimal Shadow Estimation with Minimal Measurement Settings
• Abstract summary: Shadow estimation is a powerful framework for predicting quantum properties from randomized measurements.<n>Our results establish a fundamental separation complexity: worst-case estimation requires $(d2)$ bases, whereas average-case performance requires only $(d)$ bases.
• Score: 3.398964351541323
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Shadow estimation is a powerful framework for predicting quantum properties from randomized measurements. While $3$-design protocols achieve optimal worst-case performance, the minimal number of measurement bases required for such optimality has remained open. Here we prove that $Θ(d^2)$ measurement bases are both necessary and sufficient for worst-case optimal shadow estimation and construct an explicit basis family. In stark contrast, any state $2$-design already suffices for average-case optimality: the mean squared shadow norm of normalized observables is bounded by a universal constant, and we prove strong concentration for Haar-random states, yielding constant sample complexity for generic pure-state fidelity estimation. Easily implementable $2$-designs -- from mutually unbiased bases, cyclic measurements, or shallow $\mathcal{O}(\log n)$-depth circuits -- enable optimal average-case protocols with remarkably simple measurement strategies. Our results establish a fundamental complexity separation: worst-case estimation requires $Θ(d^2)$ bases, whereas average-case performance requires only $Θ(d)$ bases, with broad implications for quantum information theory and near-term experiments.

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