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arXiv:2606.00527

Finite-Sample Selected Covariance Spectra in Classical Shadows

Masahito Hayashi

quant-phSubmitted May 30, 2026• Announced Jun 2, 2026• 0 citations
v1 · May 30, 2026v2 · Sep 22, 2026
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Abstract

We study finite-sample estimation of selected covariance matrices of classical-shadow outputs. For a general shadow-output vector, we consider its covariance matrix and a fixed selected compression. Our main theorem applies to arbitrary shadow protocols and gives an operator-norm error bound for the selected sample-centered empirical covariance. When the protocol-dependent constants appearing in this bound remain independent of the ambient system size, the required sample size is also independent of the ambient dimension. The proof combines matrix Bernstein concentration, an exact rank-one centering identity, and Weyl and Davis--Kahan perturbation bounds. We verify this bounded-output condition for local measurement settings. For general local product shadow protocols with fixed local dimension, finite-weight product observables lead to bounds controlled by support sizes and local reconstruction coefficients, not by the total number of tensor factors. Hence uniform bounds on selected set size, observable weight, and local reconstruction coefficients imply dimension-independent selected covariance estimation. For biased local Pauli shadows, we evaluate the relevant bound in closed form from the selected Pauli supports and local basis-selection probabilities. We also derive an exact covariance formula governed by Pauli compatibility and inverse-probability overlap factors, showing how measurement bias affects both diagonal variances and off-diagonal statistical couplings. A comparison with global Clifford shadows shows that this dimension-independent local behavior is not automatic for every shadow protocol.

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Real classical shadows

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Abstract Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U . In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k -local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in- k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.

Journal of Physics A Mathematical and Theoretical•Jun 2, 2025•9 citations
We study finite-sample estimation of selected covariance matrices of classical-shadow outputs. For a general shadow-output vector, we consider its covariance matrix and a fixed selected compression. Our main theorem applies to arbitrary shadow protocols and gives an operator-norm error bound for the selected sample-centered empirical covariance. When the protocol-dependent constants appearing in this bound remain independent of the ambient system size, the required sample size is also independent of the ambient dimension. The proof combines matrix Bernstein concentration, an exact rank-one centering identity, and Weyl and Davis--Kahan perturbation bounds.
We verify this bounded-output condition for local measurement settings. For general local product shadow protocols with fixed local dimension, finite-weight product observables lead to bounds controlled by support sizes and local reconstruction coefficients, not by the total number of tensor factors. Hence uniform bounds on selected set size, observable weight, and local reconstruction coefficients imply dimension-independent selected covariance estimation.
For biased local Pauli shadows, we evaluate the relevant bound in closed form from the selected Pauli supports and local basis-selection probabilities. We also derive an exact covariance formula governed by Pauli compatibility and inverse-probability overlap factors, showing how measurement bias affects both diagonal variances and off-diagonal statistical couplings. A comparison with global Clifford shadows shows that this dimension-independent local behavior is not automatic for every shadow protocol.
Classical shadows allow the prediction of many quantum-state properties from measurement data, but their efficiency strongly depends on the underlying measurement ensemble. We introduce selective GHZ measurement (SGM) shadows, which perform Greenberger-Horne-Zeilinger (GHZ)-basis measurements on randomly selected subsets of qubits and computational-basis measurements on their complement. The corresponding quantum circuits are practically feasible, with their number of CNOT gates scaling linearly with the number of qubits. This is in stark contrast to global Clifford circuits, which are generally difficult to implement. Varying the subset distribution allows tuning the SGM shadow protocol for different classes of observables with prescribed X/Y support, allowing us to outperform local Pauli as well as global Clifford shadows in certain scenarios. We determine the exact shadow channel and its inverse for arbitrary subset distributions and prove an explicit bound on the shadow norm using methods from discrete Fourier analysis and Markov semigroup theory. For natural choices of subset distributions, this gives a polynomial scaling for observables with fixed X/Y-locality. Numerical simulations of up to 50 qubits support the predicted improvements for nonlocal observables and long-range systems, and demonstrate the practical benefits of our protocol's generality by showing how the subset distributions can be tailored to settings such as fermionic modes under the Jordan-Wigner transformation.
The real classical shadows protocol of West et al. replaces the unitary (Clifford) ensemble of the Huang--Kueng--Preskill scheme by the orthogonal (real Clifford) ensemble, and for symmetric observables achieves strictly smaller estimator variances: a factor approaching two for global evolution and an exponential factor (3/2)k(3/2)^k(3/2)k for kkk-local real Pauli observables. Real hardware, however, never implements the ideal evolution. Building on the noisy classical shadows framework of Koh and Grewal, we give a complete theory of the real classical shadows protocol in the presence of a known completely positive trace-preserving noise channel acting after the orthogonal evolution. We derive the noisy global and local orthogonal shadow channels from first principles using the Weingarten calculus of the orthogonal group, prove that each is a depolarizing channel acting on the symmetric (respectively locally symmetric) component of its input, and derive from it the exact single-shot variance in closed form, together with the associated shadow seminorm, two-sided bounds on it, and the resulting sample-complexity guarantees. We prove that the noiseless sample-complexity advantages survive intact under noise. Because the variances are exact rather than bounded, the ratio is controlled by a single dimensionless parameter, which gives a closed-form criterion for when the factor of two is attainable: both the second-moment and the variance ratio reach it exactly when the observable's norm profile grows, and the noise enters that limit only through a factor lying within 2/(d+2)2/(d+2)2/(d+2) of two, so the advantage is uniform in the noise. For rank-one targets it is provably unattainable, saturating strictly below two. The local real-Pauli advantage remains (3/2)k(3/2)^k(3/2)k. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.
Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U. In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ``real'' classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords decrease the required number of samples by a factor of two. More dramatically, for k-local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in-k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.
Estimating many local expectation values over time is a central measurement bottleneck in quantum simulation and device characterization. We study the task of reconstructing the Pauli-signal matrix Sij=Tr(Oiρ(tj))S_{ij}=\text{Tr}(O_i ρ(t_j))Sij​=Tr(Oi​ρ(tj​)) for a collection of MMM low-weight Pauli observables {Oi}i=1M\{O_i\}_{i=1}^M{Oi​}i=1M​ over NNN timesteps {tj}j=1N\{t_j\}_{j=1}^N{tj​}j=1N​, while minimizing the total number of device shots. We propose a Compressed Sensing Shadow Tomography (CSST) protocol that combines two complementary reductions. First, local classical shadows reduce the observable dimension by enabling many Pauli expectation values to be estimated from the same randomized snapshots at a fixed time. Second, compressed sensing reduces the time dimension by exploiting the fact that many expectation-value traces are spectrally sparse or compressible in a unitary (e.g., Fourier) transform basis. Operationally, CSST samples m≪Nm\ll Nm≪N timesteps uniformly at random, collects shadows only at those times, and then reconstructs each length-NNN signal via standard ℓ1\ell_1ℓ1​-based recovery in the unitary transform domain. We provide end-to-end guarantees that explicitly combine shadow estimation error with compressed sensing recovery bounds. For exactly sss-sparse signals in a unitary transform basis, we show that m=O(slog⁡2slog⁡N)m=O \left(s\log^2 s \log N\right)m=O(slog2slogN) random timesteps suffice (with high probability), leading to total-shot savings scaling as Θ~(N/s)\widetildeΘ(N/s)Θ(N/s) (i.e., up to polylogarithmic factors) relative to collecting shadows at all NNN timesteps. For approximately sparse signals, the reconstruction error decomposes into a compressibility (tail) term plus a noise term. We present numerical experiments on noisy many-qubit dynamics that support strong Fourier compressibility of Pauli traces and demonstrate substantial shot reductions with accurate reconstruction.
Detecting multipartite entanglement in many qubit systems is measurement-intensive, motivating protocols that estimate only selected observables with provable efficiency. In this work we use the classical shadow protocol to study the sample complexity required to estimate a family of subsystem nnn-partite entanglement witness embedded in an larger NNN-qubit system. We derive ensemble dependent variance bounds that lead to qualitatively distinct scaling for the snapshots cost at fixed additive error εεε with numerical simulations confirm these trends, exhibiting a clear crossover from Pauli favorable performance for local witness to Clifford favorable performance as the witness becomes more global.
Abstract Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U . In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k -local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in- k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.