Classical Shadows Protocol

• Classical shadows protocol is a randomized measurement and post-processing framework that reconstructs quantum state properties from a limited number of measurement outcomes.
• It employs unitary sampling, inversion of the average measurement channel, and median-of-means estimators to provide unbiased estimates for various observables.
• The efficiency and limitations of the protocol depend on the chosen measurement ensemble, impacting sample complexity, visible operator spaces, and computational verification.

Classical shadows protocol is a randomized measurement-and-post-processing framework for predicting many properties of an unknown quantum state from a comparatively small measurement record. In its standard form, one samples a unitary $U$ from a chosen ensemble, measures $U\rho U^\dagger$ in a fixed basis, stores a classical description of the outcome, and reconstructs expectation values by inverting the associated average measurement channel. In the notation used for general basis measurements, the dephasing channel is

$$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$

and the measurement channel is

$$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$

so that a single-shot shadow state can be written as

$$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$

The protocol is organized around unbiased single-shot estimators, concentration bounds for many observables, and a choice of measurement ensemble matched to the observable class of interest (West et al., 1 Apr 2026, Fu et al., 2024).

1. Core formulation

The standard task is to estimate many linear functionals $o_i(\rho)=\operatorname{tr}(O_i\rho)$ of an unknown state $\rho$. In the measurement model emphasized across the literature, one samples a random unitary $U$, performs a computational-basis measurement, obtains an outcome $\ket{\hat b}$, and forms a classical snapshot

$$\hat\rho=\mathcal M^{-1}\!\left(U^\dagger \ket{\hat b}\!\bra{\hat b}U\right),$$

where $U\rho U^\dagger$0 is the shadow channel induced by the measurement ensemble. Repeating this $U\rho U^\dagger$1 times yields a classical shadow $U\rho U^\dagger$2, from which observable estimates are obtained through scalar random variables $U\rho U^\dagger$3 (Fu et al., 2024).

A central structural notion is the measurement channel itself. In the noiseless formulation,

$U\rho U^\dagger$4

and unbiasedness is the identity $U\rho U^\dagger$5. More generally, for group-representation-based protocols, the image of the channel,

$U\rho U^\dagger$6

is the visible space. Only observables in $U\rho U^\dagger$7 can be estimated unbiasedly. This visible-space formulation is essential once the measurement ensemble is not tomographically complete on the full operator space (Koh et al., 2020, West et al., 1 Apr 2026).

2. Statistical guarantees and estimation theory

The protocol’s basic sample-complexity guarantee is controlled by a shadow norm or an equivalent variance proxy. In the representation-theoretic formulation, the standard bound reviewed in the literature is

$U\rho U^\dagger$8

for estimating $U\rho U^\dagger$9 observables to additive error $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$0 with failure probability $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$1 (West et al., 1 Apr 2026). In the Pauli-invariant setting, this structure becomes especially explicit: if $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$2 is a Pauli observable, then its squared shadow norm is exactly $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$3, where $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$4 is the corresponding channel eigenvalue in the Pauli basis (Bu et al., 2022).

The original Huang–Kueng–Preskill analysis used a median-of-means estimator to convert variance control into logarithmic dependence on $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$5. The standard construction partitions the $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$6 snapshots into $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$7 groups, averages within each group, and returns the median of those group means. Subsequent work showed that the asymptotic constants in this post-processing layer matter operationally: the loose HKP constant $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$8 can be replaced by $$A_W(-)=\sum_w \langle w|(-)|w\rangle\,|w\rangle\langle w|,$$9 for the standard median-of-means estimator, and by $$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$0 for a modified combination-based estimator due to Minsker (Fu et al., 2024).

These refinements do not produce a uniform dominance relation. In finite-shot studies, the plain mean often had the smallest average error in benign regimes; the original median-of-means performed better than the modified estimators for Pauli measurements on Pauli-type observables; and the modified estimators performed better than standard median-of-means for global Clifford measurements of linear functions such as GHZ fidelity. The estimation layer is therefore ensemble-dependent rather than protocol-universal (Fu et al., 2024).

3. Measurement ensembles, channel structure, and visible-space geometry

The choice of unitary ensemble determines whether the shadow channel is analytically invertible, how large its eigenvalue spread is, and which observables remain visible. A broad structural result is that for any Pauli-invariant unitary ensemble,

$$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$1

so the channel is diagonal in the Pauli basis and inversion is immediate whenever all eigenvalues are positive (Bu et al., 2022). This framework subsumes global Clifford, local Clifford, and locally scrambled ensembles, and it expresses sample complexity directly in terms of the inverse channel spectrum.

A complementary generalization uses arbitrary compact-group representations. For a basis $$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$2, the key notion is a centralizing basis, for which the measurement channel acts as a scalar on each visible isotypic component. A non-degenerate commuting subgroup eigenbasis yields

$$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$3

with $$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$4 the dimension of the $$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$5-invariant subspace in the $$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$6-irrep. This extends analytic inversion beyond multiplicity-free settings and reduces classical post-processing to sectorwise rescaling (West et al., 1 Apr 2026).

The same visible-space logic explains why some protocols deliberately trade universality for improved variance. In locally entangled shadows, Bell-basis measurements on qubit pairs give

$$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$7

for dimer-compatible Pauli strings of weight $$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$8, improving the Pauli-shadow scaling $$M=\mathbb E_{U\sim E}\,{\rm Ad}_{U^\dagger}\circ A_W\circ {\rm Ad}_U, \qquad {\rm Ad}_U(-)=U(-)U^\dagger,$$9, but incompatible Pauli operators become unlearnable because the corresponding channel eigenvalues vanish. For $$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$0-qubit GHZ-basis measurements on blocks, the best-case compatible-operator scaling approaches $$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$1, again on a progressively smaller observable class (Ippoliti, 2023).

4. Symmetry-adapted and hardware-specific protocols

A major direction in the literature is to tailor the randomized measurement ensemble to symmetry sectors or to a hardware-native transformation group. Real classical shadows replace unitary Cliffords by orthogonal Cliffords and use a real measurement basis. In the global orthogonal case,

$$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$2

so the visible space is the real-symmetric operator sector. For arbitrary real-valued observables this reduces the required number of samples by a factor of about $$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$3 asymptotically, while in the local orthogonal case a weight-$$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$4 Pauli string over $$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$5 has variance bounded by $$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$6, improving on the $$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$7 local-unitary bound (West et al., 2024).

Fermionic and matchgate shadows adapt the protocol to Majorana observables and fermionic Gaussian unitaries. The unified matchgate framework proved that the continuous $$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$8 ensemble, its Clifford intersection, the $$\hat\rho=M^{-1}\!\left(U^\dagger \Pi_w U\right), \qquad \Pi_w=|w\rangle\langle w|.$$9-based constructions, and perfect-matching-based sub-ensembles are equivalent at the level of the first three moments relevant for classical shadows. It then derived a smaller sub-ensemble of matchgate circuits that is optimal in terms of number of gates while inheriting the same performance guarantees (Heyraud et al., 2024).

For lattice gauge theories, symmetry-aware protocols exploit the physical Hilbert space rather than the full unconstrained link Hilbert space. In a $o_i(\rho)=\operatorname{tr}(O_i\rho)$0 lattice gauge theory, the Global Dual Pairs, Local Dual Pairs, and Dual Product protocols estimate gauge-invariant observables in the dual variables and can offer exponential improvements in sample complexity over symmetry-agnostic product shadows, at the cost of increased circuit depth and string-like mapped operations (Bringewatt et al., 4 Nov 2025). In photonic systems, randomized passive linear optical transformations combined with photon-number measurements define a sectorwise shadow protocol on fixed total photon-number spaces. The resulting estimator is efficient for low-degree observables of interest, but the protocol does not access coherences between different photon-number sectors (Thomas et al., 8 Oct 2025).

5. Robustness and task-specific generalizations

A substantial body of work modifies the protocol to handle nonideal devices or to target quantities other than $o_i(\rho)=\operatorname{tr}(O_i\rho)$1 for a fixed mixed state. For known noise channels $o_i(\rho)=\operatorname{tr}(O_i\rho)$2, the correct estimator is obtained by inverting the noisy shadow channel

$o_i(\rho)=\operatorname{tr}(O_i\rho)$3

not the noiseless one. This preserves unbiasedness under noise and yields sample-complexity bounds in terms of a noisy shadow seminorm. For unitary 2-designs, the noisy channel collapses to an effective depolarizing channel, making the correction explicit for depolarizing noise and amplitude damping (Koh et al., 2020). For noisy readout with crosstalk, $o_i(\rho)=\operatorname{tr}(O_i\rho)$4-twirling symmetrizes the readout channel into a translation-invariant form on bit strings, diagonal in the Boolean Fourier basis, so mitigation reduces to scalar corrections by Fourier coefficients $o_i(\rho)=\operatorname{tr}(O_i\rho)$5 rather than inversion of a full confusion matrix (Nguyen, 2023).

The i.i.d. assumption can also be removed. For adaptive or history-dependent sequences of states $o_i(\rho)=\operatorname{tr}(O_i\rho)$6, one can keep the usual single-shot shadow values $o_i(\rho)=\operatorname{tr}(O_i\rho)$7 but replace empirical averaging by a truncated mean

$o_i(\rho)=\operatorname{tr}(O_i\rho)$8

Using conditional unbiasedness and Freedman’s inequality for martingales, this yields the same shadow-norm scaling for estimating the time-averaged observable $o_i(\rho)=\operatorname{tr}(O_i\rho)$9 even under arbitrary temporal correlations (Zambrano, 5 Mar 2026).

The target of the protocol can also be changed. Principal eigenstate classical shadows assume $\rho$0 has a dominant eigenstate $\rho$1 with eigenvalue $\rho$2 and use a collective symmetric measurement on $\rho$3 copies, together with the affine estimator

$\rho$4

to estimate $\rho$5 rather than $\rho$6. In the regime $\rho$7, the sample complexity matches the optimal pure-state rate (Grier et al., 2024). Classical shadows have also been used to prove $\rho$8-copy approximation guarantees in a local quantum Wasserstein-1 distance (Palma et al., 2023) and to build hypothesis-testing verification protocols such as DPSO, whose sample complexity is

$\rho$9

for a level-$U$0 verification strategy operator $U$1 (Li, 2024).

6. Limitations, verification complexity, and conceptual scope

A recurring misconception is that classical shadows are universally tomographically complete. In fact, completeness depends on the ensemble and basis. Bell shadows lose all operators that cut a dimer; local orthogonal shadows restrict visibility to $U$2; photonic passive-linear-optics shadows lose inter-photon-number coherences; fermionic matchgate shadows are naturally tied to parity-preserving fermionic sectors; and in the compact-group framework invisibility occurs exactly when an irrep has no subgroup-invariant component, $U$3 (Ippoliti, 2023, West et al., 2024, Thomas et al., 8 Oct 2025, West et al., 1 Apr 2026).

A second limitation is computational rather than statistical. The existence of a concise shadow does not imply that arbitrary purported shadow data are easy to certify. The computational problem of classical shadow validity—whether a reported shadow is consistent with some physical quantum state—is QMA-complete even for the standard local-Clifford protocol of Huang, Kueng, and Preskill, and for exponentially many observables the corresponding consistency problem is complete for the class $U$4, a quantum-classical analogue of the second level of the polynomial hierarchy (Karaiskos et al., 9 Oct 2025). Efficient prediction from trusted shadow data and efficient verification of arbitrary shadow reports are therefore distinct tasks.

Taken together, these developments define classical shadows less as a single protocol than as a design paradigm. The common backbone is randomized measurement, inversion of an average measurement channel, and reuse of one measurement record for many later queries. The modern literature extends that backbone to arbitrary group representations, symmetry-reduced sectors, noisy and non-i.i.d. experiments, collective measurements, and hardware-specific settings, while repeatedly emphasizing the same structural principle: sample efficiency depends on matching the measurement ensemble to the observable sector one actually needs to predict (West et al., 1 Apr 2026).