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arXiv:2606.20003

Optimal Shadow Estimation with Minimal Measurement Settings

Zhiyao Yang, Datong Chen, Huangjun Zhu

quant-phmath-phSubmitted Jun 18, 2026• Announced Jun 19, 2026
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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.

Comments
8+21 pages and 3+5 figures; comments and suggestions are very welcome!

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Shadow estimation is a powerful framework for predicting quantum properties from randomized measurements. While 333-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(d2)Θ(d^2)Θ(d2) measurement bases are both necessary and sufficient for worst-case optimal shadow estimation and construct an explicit basis family. In stark contrast, any state 222-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 222-designs -- from mutually unbiased bases, cyclic measurements, or shallow O(log⁡n)\mathcal{O}(\log n)O(logn)-depth circuits -- enable optimal average-case protocols with remarkably simple measurement strategies. Our results establish a fundamental complexity separation: worst-case estimation requires Θ(d2)Θ(d^2)Θ(d2) bases, whereas average-case performance requires only Θ(d)Θ(d)Θ(d) bases, with broad implications for quantum information theory and near-term experiments.
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Approximate unitary designs reproduce the statistics of Haar-random unitaries to a given order and accuracy. Recent constructions realize such designs with logarithmic-depth circuits, enabling shallow measurement protocols for various quantum state-learning tasks while preserving performance. These results raise the question of whether approximate designs can replace exact designs more generally. We show that even exponentially small design error need not preserve the learning guarantees of exact designs. This motivates deriving learning guarantees directly from the measurement circuit structure. For observable estimation with classical shadows, we prove that logarithmic-depth two-layer Clifford circuits yield an unbiased estimator matching the global Clifford variance scaling for every state and Hermitian observable. Beyond observable estimation, this bound allows shallow measurements to retain global Clifford guarantees for other tasks, including setting-efficient tomography, mixed-state metrology, and stabilizer structure learning. We complement these statistical guarantees with a compact, exact tensor-network representation of the inverse shadow channel. When measurement bases are reused, approximate unitary designs of arbitrarily high order need not uniformly match the Haar variance in the many-shot limit. For the all-to-all random two-local circuits considered, doing so requires nearly linear depth. Together, our results reveal the capabilities and limitations of shallow quantum state learning, highlighting the distinction between approximating Haar randomness and reproducing its learning guarantees.
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We give a shadow tomography algorithm that uses O(log⁡(m)log⁡(m/δ)/ε2)O(\log(m) \log(m/δ) / \varepsilon^2)O(log(m)log(m/δ)/ε2) copies of the input state. Specifically, given a quantum state ρρρ and observables O1,…,OmO_1,\ldots,O_mO1​,…,Om​, the algorithm outputs estimates of tr(Oiρ)\mathrm{tr}(O_iρ)tr(Oi​ρ) for all i∈[m]i\in[m]i∈[m] that are all accurate to within ±ε\pm\varepsilon±ε, except with probability at most δδδ. The key idea is to work in the average-case setting: for a random state drawn from a known prior, a pretty good measurement on O(log⁡(m)/ε2)O(\log(m)/\varepsilon^2)O(log(m)/ε2) copies gives a constant-error guarantee on all but a small fraction of the prior. Boosting this guarantee gives an average-case shadow tomography algorithm, and the minimax principle then (non-constructively) implies the existence of a measurement with the same worst-case guarantee. We also identify an obstruction to further improvement with this approach. While pretty good measurements amplify success over the randomness of the prior, they do not amplify success over the randomness of measurement. This issue appears to be inherent, and gives rise to the remaining log⁡(m)\log(m)log(m) factor separating the complexity of the quantum problem from its classical counterpart.
We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown ddd-dimensional quantum state ρρρ and a known set of observables {Oi}i=1m\{O_i\}_{i=1}^m{Oi​}i=1m​, the goal is to estimate expectation values {tr(Oiρ)}i=1m\{\mathrm{tr}(O_iρ)\}_{i=1}^m{tr(Oi​ρ)}i=1m​ to accuracy εεε in LpL_pLp​-norm, using possibly adaptive measurements that act on O(polylog(d))O(\mathrm{polylog}(d))O(polylog(d)) number of copies of ρρρ at a time. We focus on the regime where εεε is below an instance-dependent threshold. Our main contribution is an instance-optimal characterization of the sample complexity as Θ~(Γp/ε2)\tildeΘ(Γ_p/ε^2)Θ~(Γp​/ε2), where ΓpΓ_pΓp​ is a function of {Oi}i=1m\{O_i\}_{i=1}^m{Oi​}i=1m​ 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∞L_\inftyL∞​-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form ∑i=1mαiOi\sum_{i=1}^m α_i O_i∑i=1m​αi​Oi​ with ∥α∥q=1\|α\|_q = 1∥α∥q​=1 (where qqq is dual to ppp) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of Θ(Γpob/ε2)Θ(Γ^{\mathrm{ob}}_p/ε^2)Θ(Γpob​/ε2). We then show Θ~(Γp/ε2)\tildeΘ(Γ_p/ε^2)Θ~(Γp​/ε2) 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∞ob=Γ∞Γ^{\mathrm{ob}}_\infty = Γ_\inftyΓ∞ob​=Γ∞​. In both cases, allowing ccc-copy measurements improves the sample complexity by at most Ω(1/c)Ω(1/c)Ω(1/c). Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.