Classical Shadows With Noise

1.1.1 Review of Classical Shadows

• Classical shadows sample a unitary, measure in the computational basis, and invert the shadow channel to form unbiased state estimates for predicting linear observables.Median-of-means estimation and a shadow-norm variance bound yield state-independent sample-complexity guarantees [11].Source: §1.1.1 · p. 4

1.1.2 Our Contributions

• To keep noisy classical shadows unbiased, the protocol must apply a noise-dependent inverse shadow channel during classical post-processing.The noisy shadow channel and its inverse depend on the error channel, assuming the inverse exists.Source: §1.1.2 · pp. 5–6
• With noise, classical shadows still admit sample-complexity upper bounds for estimating many observables, with only polynomial-factor increases in sample count.The bounds cover general noise channels and are expressed using a shadow seminorm.Source: §1 · p. 3, §1.1.2 · p. 6
• For Pauli observables, the paper derives noisy sample-complexity bounds under tensor-product single-qubit noise, assuming identical channels for simplicity.The result generalizes to different single-qubit noise channels; identity noise recovers the noiseless bound [11].Source: §1.1.2 · p. 6
• The paper also establishes noisy-channel invertibility conditions, tensor-product factorization, a depolarizing shadow channel for 2-designs, and applications to depolarizing noise and amplitude damping.For general noise, the generalized shadow norm retains seminorm properties; the results yield rigorous sample-complexity upper bounds for both noise models.Source: §1.1.2 · p. 7

1.2.1 Property Estimation and Quantum Tomography

• State-estimation alternatives include full, matrix-product-state, MERA, neural-network, and overlapping tomography, while shadow tomography estimates many observables and classical shadows apply when shadow norms are small.Full tomography includes sample-optimal protocols requiring entangled measurements and more experimentally friendly single-sample methods.Source: §1.2.1 · pp. 7–8

1.2.2 Solving the Measurement Problem

• Approaches to the near-term measurement problem include Pauli grouping [40] [41], unitary partitioning [42] [43], engineered likelihood functions [44], and deep learning [35].The paper directs readers to [11] for comparisons with classical shadows.Source: §1.2.2 · p. 8

1.2.3 Classical Shadows

• Related classical-shadow work covers fermionic extensions [47] [48] [49], molecular-energy estimation, entanglement detection [51], and an optical experiment; Chen et al. [18] study noise without assuming its model is known beforehand.This paper assumes the noise has been modeled before protocol implementation; later extensions and applications are also noted [48, 49, 55–90].Source: §1.2.3 · p. 8

2 Mathematical Preliminaries

• The preliminaries fix notation for operator, Hermitian, unitary, and density-operator spaces, Haar measure, spectral norm, integer sets, and generalized Kronecker delta.Source: §2 · p. 9

2.1 Linear Superoperators and Quantum Channels

• A quantum channel is completely positive and trace-preserving, and the section defines completely dephasing, depolarizing, and amplitude damping channels.Source: §2.1 · pp. 9–11

2.2 t-Fold Twirls and t-Designs

• A t-design reproduces the Haar-random t-fold twirl; the n-qubit Clifford group is a 3-design but not a 4-design [16] [17].Source: §2.2 · p. 12
• The section states a useful identity for a qudit 3-design, a linear superoperator, and three linear operators, for use later in the paper.Source: §2.2 · p. 12

3.1 Generating the Classical Shadow

• With an invertible shadow channel, inverse-corrected noisy shadows reproduce the input state in expectation, so they give unbiased estimates of linear observables.Each measurement applies a random unitary, the noise channel, and a computational-basis measurement; median-of-means is used for estimation.Source: §3.1 · p. 13, §3.2 · p. 14

3.2 Noisy Classical Shadows Protocol

• The shadow seminorm yields a variance bound independent of the unknown state, and the protocol gives a sample-complexity guarantee for estimating multiple observables.The guarantee assumes an invertible shadow channel and controls accuracy and failure probability using the observables’ shadow seminorms.Source: §3.2 · pp. 14–15

3.3 Product Ensembles with Product Noise

• For product unitary ensembles and product noise channels, the shadow channel, its inverse, and the classical shadow factorize into tensor products.This factorization holds when each single-qubit shadow channel is invertible, so product noise preserves the protocol’s tensor-product structure.Source: §3.3 · p. 17
• For a k-local observable under product ensembles and product noise, the shadow seminorm reduces to a bound determined by the factors on its k supported qubits.The result assumes an invertible shadow channel; for the identity observable (k = 0), the seminorm is 1.Source: §3.3 · pp. 18–19

4.1 Derivation of Shadow Channel

• The Clifford shadow channel is depolarizing for any quantum channel, and it is invertible exactly when tr(E ◦ diag) ≠ 1; noise cannot increase its depolarizing parameter above the noiseless value.The depolarizing parameter can also be negative for some channels.Source: §4.1 · pp. 22–23

4.2 Classical Shadow

• When the shadow channel is invertible, the paper gives an explicit classical-shadow estimator by applying the inverse depolarizing channel to the measurement outcome.The estimator is stated for an n-qubit 2-design and a quantum channel.Source: §4.2 · p. 23

4.3 Derivation of Shadow Seminorm

• For traceless observables, the paper derives shadow-seminorm expressions and bounds that yield sample-complexity guarantees for Clifford ensembles and other unitary 3-designs.The sample-complexity result applies to estimating a collection of linear target functions, assuming tr(E ◦ diag) ≠ 1.Source: §4.3 · pp. 24–25

4.4.1 Noiseless Case

• In the noiseless case, the estimator is (2^n + 1) U†|b⟩⟨b|U − I, and the same results hold under dephasing noise.Source: §4.4.1 · p. 26

4.4.2 Depolarizing Channel

• Under depolarizing noise with parameter f ∈ [0, 1], the shadow-channel parameter is f/(2^n + 1), and the shadow seminorm is bounded by 3/f^2 tr(O^2).A sample-complexity bound is also derived for estimating M observables within error ε and failure probability δ.Source: §4.4.2 · pp. 26–27

4.4.3 Amplitude Damping Channel

• For amplitude-damping noise with parameter p, the paper derives the shadow channel, inverse, estimator, seminorm, and a sample-complexity bound for a collection of observables.The bound depends on p, n, ε, and the largest tr(O_i^2).Source: §4.4.3 · pp. 28–29

5.1 Derivation of Shadow Channel

• For product ensembles with single-qubit 2-designs and a single-qubit channel E, the section derives the shadow channel and its inverse, which is invertible when tr(E ◦ diag) ≠ 1.The product Clifford ensemble is an example of the product 3-design ensembles covered by the section.Source: §5.1 · p. 29, §5 · p. 29

5.2 Classical Shadow

• For the same product setting, when tr(E ◦ diag) ≠ 1, the section derives an expression for the classical shadow.The ensemble and outcome are written as tensor products of single-qubit elements and bit values.Source: §5.2 · p. 29

5.3 Derivation of Shadow Seminorm

• The section derives shadow seminorms for k-local Pauli observables and uses them to bound sample complexity for M such targets under E^⊗n.For general product-channel noise, the authors report no simple expression for a general k-local observable; Appendix D gives a bound for depolarizing noise.Source: §5.3 · pp. 30–31

5.4 Examples

• The examples apply the results to identity, depolarizing, and amplitude-damping channels, deriving shadow expressions and k-local Pauli seminorm bounds that imply sample-complexity bounds.The noise channel is fixed in each example, and the product Clifford ensemble is used.Source: §5.4 · p. 31

5.4.1 Noiseless Channel

• In the noiseless local case, f(I) = 1/3, the k-local Pauli seminorm is 3^wt(P), and the sample-complexity bound follows from Corollary 5.5.These results recover Eqs. (S44) through (S50) in [11].Source: §5.4.1 · p. 32

5.4.2 Depolarizing Channel

• For depolarizing noise, f(D1,f) = f/3; the section derives shadow expressions, with the sample-complexity bound following from Corollary 5.5.The expressions are obtained by applying Claims 5.1 and 5.2.Source: §5.4.2 · p. 32

5.4.3 Amplitude Damping Channel

• For n=1 under local amplitude damping, f(AD1,p) = p/3; Claim 5.1 gives expressions for the shadow channel and inverse, and sample complexity in this setting follows from Corollary 5.5.Source: §5.4.3 · pp. 32–33

Concluding remarks

• For depolarizing, dephasing, and amplitude-damping noise, the authors derive shadow-seminorm sample bounds and modify post-processing to obtain an estimator that remains unbiased under noise.They conclude that classical shadows can remain efficient for certain estimation tasks despite noise.Source: §6 · p. 33

Open problems

• The authors leave open whether a noisy-shadow estimator remains useful when the assumed channel F differs from the true channel E, and how their approach compares with [18].They ask whether their approach is preferable when F and E are sufficiently close.Source: §6 · p. 33
• The GTM assumption models noise as a channel independent of the unitary and computation time, but the authors call for analysis of more realistic gate-dependent or non-Markovian noise.They note that accounting for noise both before and after a unitary requires only a minor modification to Algorithm 2’s quantities.Source: §6 · pp. 33–34
• An experimental demonstration remains open: show that inverting the noisy rather than noiseless shadow channel improves estimation accuracy.The authors point to prior classical-shadows experiments [66] [79] as possible foundations.Source: §6 · p. 34
• The authors ask for simple necessary and sufficient conditions for invertibility of the noisy shadow channel, extending existing sufficient and ensemble-specific results.The cited results concern tomographic completeness for noiseless shadows and a 2-design condition involving tr(E ◦ diag) ≠ 1.Source: §6 · p. 34
• The authors call for theoretical, numerical, or experimental comparisons of classical shadows with competing methods on noisy devices, including whether advantages for molecular-Hamiltonian estimation persist under noise.The cited comparison of locally-biased shadows with grouping and ℓ1-sampling found an advantage for molecular Hamiltonians.Source: §6 · p. 34

A.1 Proof of Lemma 2.6

• Claim A.1 states a property of the unary operator ()‡ defined in Eq. (47).The supplied passage does not include the property’s displayed formula.Source: §A.1 · p. 35
• For t = 3, the Haar integral is expressed using R+ and R− with dimension-dependent coefficients and traces against those operators.The proof introduces permutation operators for S3 and linear combinations of them to evaluate this integral.Source: §A.1 · pp. 35–36
• Lemma A.2 gives third-order integral identities for |zxx⟩⟨yxx| and Γ ⊗ |xx⟩⟨xx| in terms of R+, R0, and R1.The expressions depend on Kronecker-delta terms for the first operator and on tr(Γ) plus ⟨x|Γ|x⟩ for the second.Source: §A.1 · p. 36
• Setting Γ = Λ(|x⟩⟨x|) yields a third-order integral involving tΛ,x and Λxxxx, which supports the identity used in the proof.The passage identifies tr(Γ) = tΛ,x and ⟨x|Γ|x⟩ = Λxxxx.Source: §A.1 · p. 37
• Expanding R+, R0, and R1 converts permutation-operator expressions into trace combinations involving B and C.The displayed combinations include tr(B)tr(C), tr(BC), B tr(C), C tr(B), BC, and CB.Source: §A.1 · pp. 37–38
• The proof applies a qudit 3-design and linear superoperator E to operators A, B, and C to establish Lemma 2.6.It uses the 3-design assumption for U† and the identity relating trace preservation of E to unitality of E‡.Source: §A.1 · pp. 38–39
• The final expression is organized by α = tr(E(I)) and β = tr(E ◦diag), with terms formed from traces of A, B, and C and their products.The displayed terms include tr(A)tr(BC), tr(AB)tr(C), tr(ABC), and tr(ACB).Source: §A.1 · p. 39

• Because tr(ρ) = tr(ˆρ) = 1, the centered estimator equals tr(Ooˆρ) = tr(Ooρ), so its variance depends only on the traceless part of O.Source: §A.2 · p. 39
• To obtain an a priori variance bound independent of ρ, the proof maximizes over all quantum states.Source: §A.2 · p. 40

A.3 Proof of Corollary 4.6

• For A ∈ H_2^n, 2^-n tr(A) ≤ ∥A∥sp ≤ tr(A).Source: §A.3 · p. 40
• The proof first addresses the lower bound.Source: §A.3 · p. 40
• The proof then addresses a first upper bound and a second upper bound.Source: §A.3 · p. 41

B When is the Shadow Seminorm a Norm?

• The shadow seminorm is a seminorm, and under the stated sufficient conditions it is point-separating and therefore a norm; depolarizing, dephasing, and amplitude damping channels qualify.The appendix states these channels satisfy the conditions for the shadow seminorm to be a norm.Source: B When is the Shadow Seminorm a Norm? · pp. 41, 43
• The sufficient channel condition is nontrivial: although every unital channel belongs to Λ_n, there exist quantum channels outside Λ_n, so the condition does not include all channels.Source: B When is the Shadow Seminorm a Norm? · p. 42
• The seminorm property follows by verifying the triangle inequality and absolute homogeneity.The proof handles the triangle inequality directly and obtains absolute homogeneity from linearity.Source: B When is the Shadow Seminorm a Norm? · pp. 42–43

C Inconsequential Noise

• For an n-qubit 2-design, a linear superoperator E leaves the shadow channel unchanged if and only if tr(E ◦ diag) = 2^n.This section characterizes such channels as inconsequential noise.Source: C Inconsequential Noise · p. 44
• Equivalently, E is inconsequential exactly when ⟨b|E(|b⟩⟨b|)|b⟩ = 1 for every b ∈ {0, 1}^n.Source: C Inconsequential Noise · p. 44

D Shadow Seminorm of k-Local Observable with Product Clifford Ensemble

• The seminorm analysis treats a k-local observable under depolarizing noise with the product Clifford ensemble, rather than a general quantum channel.Source: D Shadow Seminorm of k-Local… · p. 44
• For the observable’s Pauli-basis expansion and 0 ≤ f ≤ 1, Proposition D.1 gives a casewise factor of 1/f when p = q = 0.Source: D Shadow Seminorm of k-Local… · p. 44
• The casewise factor is 1 when exactly one of p or q is zero, and 3/f^2 when p = q ≠ 0.Source: D Shadow Seminorm of k-Local… · p. 44
• The casewise factor is 0 in all other p, q cases.Source: D Shadow Seminorm of k-Local… · p. 44

E Noisy Input States

• The noisy shadow channel, classical shadow, and shadow seminorm are modified to account for the input noise channel K.Source: E Noisy Input States · pp. 48–49
• The extension assumes both the shadow channel and K are invertible linear superoperators, without requiring their inverses to be quantum channels.Source: E Noisy Input States · p. 48
• Theorem 3.7 carries over with the shadow seminorm ∥·∥shadow,U,E replaced by ∥·∥shadow,U,E,K.The appendix follows the same argument as in the main text.Source: E Noisy Input States · p. 49
• Algorithm 2 uses the modified shadow seminorm in step 2 and replaces each M^-1_U,E occurrence in step 8 with K^-1 ◦ M^-1.Source: E Noisy Input States · p. 49