Real Classical Shadows with Noise

Abstract: 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 $(3/2)<sup>k$ for $k$-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 $2/(d+2)$ 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 $(3/2)<sup>k$. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.

• The paper develops a complete noisy orthogonal-shadow theory showing that known CPTP noise changes the shadow channel only through a scalar parameter β, preserving unbiased inversion and the protocol’s depolarizing structure.
• Real shadows retain a near-twofold advantage for global observables and an exact (3/2)^k advantage for k-local Pauli observables under arbitrary product noise, while rank-one targets generally fall short of the factor of two.
• The results identify important design limits: real bases cannot estimate antisymmetric observables, complex bases expand the visible space at a variance cost, and the theory assumes known, stationary, gate-independent Markovian noise.

The paper develops a complete theory of classical shadows estimation under known noise when the measurement ensemble is the orthogonal group rather than the unitary group, extending the real (orthogonal) classical shadows protocol of West et al. to the noisy setting previously treated only for Clifford ensembles by Koh and Grewal. The central finding is that the sample-complexity advantages of real shadows — a factor approaching two for global observables and $(3/2)^k$ for $k$-local Pauli observables — survive arbitrary completely positive trace-preserving (CPTP) noise intact, because the noise enters the shadow channel only through a single scalar that rescales a depolarizing parameter without altering the channel's structure.

Setting and noise model

The protocol estimates $\tr(O\rho)$ by evolving $\rho$ with a random orthogonal $U \in O(d)$ ($d = 2^n$), measuring in a fixed basis, and classically post-processing the back-evolved projector. The ensemble is realized in practice by real Clifford circuits via their orthogonal 3-design property. The noise model follows Koh and Grewal: a known CPTP channel $E$ acts once between evolution and measurement. The key scalar invariant is the diagonal weight

$\beta = \tr[E \circ diag] = \sum_b \bra b E(\Pi_b)\ket b,$

with $1 < \beta \le d$ for any non-trivial CPTP channel. A companion scalar $\widetilde\beta$, defined via a transposed dephasing map, governs complex-basis settings.

The noisy global orthogonal shadow channel

Using the order-two orthogonal Weingarten calculus, the paper derives the noisy shadow channel in closed form: for trace-preserving or unital $k$0,

$k$1

a depolarizing channel acting on the symmetric component of its input. Inverting on the symmetric subspace restores unbiasedness whenever $k$2. Notably, the derivation never invokes any reality or conjugation-symmetry property of the noise: coherent over-rotations with complex Kraus operators are absorbed entirely into $k$3, and readout error enters through $k$4. Dephasing noise is provably inconsequential ($k$5). Noise-blind post-processing incurs an exact multiplicative bias $k$6, which cannot be removed by averaging.

Exact variance and the factor-of-two criterion

The shadow seminorm is computed exactly via the order-three Weingarten function, yielding closed-form single-shot variances and a sample-complexity bound with prefactor 170 (versus 204 for the unitary case). More importantly, the exact second-moment ratio between unitary and orthogonal protocols depends on the state and observable through a single dimensionless parameter

$k$7

via the master identity $k$8, which tends to $k$9 at large $\tr(O\rho)$0. Since the noise enters only through $\tr(O\rho)$1, whose deviation from two is bounded by $\tr(O\rho)$2 uniformly in $\tr(O\rho)$3, the factor of two is attained as $\tr(O\rho)$4 uniformly in the noise — a strong robustness claim. Conversely, for rank-one targets $\tr(O\rho)$5 saturates at $\tr(O\rho)$6 and the ratio provably stays below two (e.g., $\tr(O\rho)$7 at depolarizing strength $\tr(O\rho)$8), so the factor of two is unattainable there regardless of system size.

Local ensemble and the exponential advantage

For the product ensemble $\tr(O\rho)$9 with product noise, the channel factorizes across qubits, and the locally symmetric Pauli seminorm equals $\rho$0 where $\rho$1 depends on the single-qubit noise only through one entry of its Pauli transfer matrix. Because $\rho$2 independent of the noise, the local real-Pauli advantage remains exactly $\rho$3 under arbitrary product CPTP noise.

Complex bases and the enlarged visible space

Introducing bases of arbitrary reality fraction $\rho$4, the channel becomes $\rho$5 with $\rho$6. Whenever the basis is not real, the visible space enlarges from the symmetric subspace to all of $\rho$7, making antisymmetric (time-reversal-odd) observables estimable. The unitary protocol is recovered in the joint limit $\rho$8, $\rho$9: the depolarizing parameters agree to relative error $U \in O(d)$0, but the second moment approaches its unitary counterpart from opposite directions depending on whether the observable is dominated by $U \in O(d)$1 or $U \in O(d)$2, so at small $U \in O(d)$3 a fully complex basis can be markedly worse than unitary shadows (a factor $U \in O(d)$4 at $U \in O(d)$5 in the worked example).

Many-body case studies

Three regimes illustrate the criterion. For GHZ fidelity (rank-one), exact ratios converge to $U \in O(d)$6–$U \in O(d)$7, well short of two. For the critical transverse-field Ising Hamiltonian in its ground state, $U \in O(d)$8 grows exponentially, and the variance ratio reaches $U \in O(d)$9 by ten qubits — essentially the full factor of two. The scalar spin chirality, being antisymmetric, lies in the kernel of the real-basis channel for every CPTP $d = 2^n$0: no amount of data recovers it, and a complex basis is required, at a quantified price (a 38% variance increase for the visible energy observable at intermediate reality).

Design-independence and the Brauer machinery

All quantities depend on the ensemble only through moments of order at most three, so every orthogonal 3-design — including the finite real Clifford group — yields literally identical predictions. The paper also shows the 3-design is not variance-optimal among 2-designs: a subgroup $d = 2^n$1 reproduces the shadow channel exactly while beating every 3-design by 20.6% on a particular observable. An extended appendix develops the Brauer algebra and proves that although the order-three Gram matrix is singular at $d = 2^n$2 (rank dropping from 15 to 10, with five explicit null vectors tied to the Levi-Civita Fierz identity), the physical linear systems remain solvable there, with coefficients finite because the denominator $d = 2^n$3 excludes the factor $d = 2^n$4.

Limitations and open questions

The theory assumes gate-independent, time-stationary, Markovian noise acting once after the ideal evolution; layered or gate-dependent noise produces observable-dependent bias not captured by any single scalar. Inversion requires the channel to be characterized in advance; the cost of calibrating $d = 2^n$5 to given accuracy for $d = 2^n$6 is left open. Whether the orthogonal 3-design is minimax-optimal among 2-designs, whether logarithmic-depth approximate orthogonal designs exist, and how the collapse onto $d = 2^n$7 degrades at shallow circuit depth are all stated as open problems. The seminorm bounds themselves are loose (the measured median-of-means guarantee overshoots by roughly a factor of 40 in shots, attributable to the Chebyshev constant rather than the norm relaxation).

Conclusion

This work establishes that noisy real classical shadows inherit the full structural theory of their unitary counterparts, with the orthogonal Weingarten calculus replacing the unitary one. The practical message is precise: the global factor-of-two and local $d = 2^n$8 advantages are properties of the orthogonal 3-design class as a whole, uniform in known CPTP noise, governed by a single dimensionless parameter set by the observable's norm profile, and forfeited only for rank-one targets or time-reversal-odd observables requiring complex bases.