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REVIEW 3 major objections 6 minor 35 references

Quantum Spectral Anomaly Detection

T0 review · 3 major / 6 minor · reviewed 2026-07-07 · glm-5.2

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Pith's one-line read Smooth spectral threshold replaces hard PCA cutoff for quantum anomaly detection

desk verdict QSPADE derives a soft spectral anomaly detector from a variational principle and gives it a measurement-based quantum implementation. The math is clean; the main gap is that the dimension-independent sample complexity counts only measurement shots, not the per-shot cost of Hamiltonian simulation. read the letter →

arxiv 2607.05307 v1 pith:3L6OYGAT submitted 2026-07-06 quant-ph cs.LG

Yewei Yuan , Michele Minervini , Mark M. Wilde , Nana Liu This is my paper · ORCID
classification quant-phcs.LG
keywords quantumanomalydetectionprincipalcomponentanalysisspectralmethodssoftthresholdingmeasurementHamiltoniansimulationkernelphasetransitions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces QSPADE (Quantum Spectral Anomaly Detection), a method that computes PCA-like anomaly scores directly from the eigenvalue spectrum of the average quantum state of a normal dataset, without recovering eigenvectors or constructing Gram matrices. The central object is the soft spectral detector M^f_{μ,T} = f((C − μI)/T), where C is the average normal state, f is a smooth monotone response function, μ is a spectral threshold, and T is a temperature parameter controlling how sharply the response transitions. This detector replaces the binary accept-or-reject rank cutoff of standard PCA with a graded threshold: eigenvalues well above μ are fully accepted, those well below are fully rejected, and borderline components contribute fractionally. As T approaches zero, the soft detector converges to the hard spectral projector used in standard PCA. The threshold μ is calibrated by matching the retained spectral mass to a target fraction α, rather than choosing a discrete rank K. The paper provides two quantum implementations—a qumode circuit that sweeps the full spectral threshold distribution and a qubit Hadamard-test circuit for single-target calibration—and shows that the residual Q score can be calibrated with sample complexity O(η⁻² log δ⁻¹), independent of the Hilbert space dimension d. Numerical experiments demonstrate that QSPADE reproduces kernel-PCA-like nonlinear decision boundaries on encoded classical data and tracks the quantum phase transition in a transverse-field Ising model without specifying an order parameter.

What carries the argument

Soft spectral detector M^f_{μ,T} = f((C − μI)/T); retained-mass calibration R^f_{C,T}(μ) = α; qumode position-quadrature measurement implementing f via a reflected density g_{f,T₁}; qubit Hadamard-test estimator representing f as a superposition of phase functions e^{itx}; Dvoretzky–Kiefer–Wolfowitz inequality for dimension-independent calibration; Robbins–Monro stochastic approximation for single-target threshold search.

What would settle it

If the cost of implementing the controlled unitary e^{itωC/T} (or the qumode interaction e^{-ip̂⊗C/T₂}) scales with the Hilbert space dimension d in a way that dominates the O(η⁻² log δ⁻¹) measurement-shot count, then the end-to-end complexity of QSPADE calibration is not dimension-independent, and the efficiency advantage over classical PCA vanishes for large d.

Watch

Extended reading notes

Core claim

The core discovery is that PCA-style anomaly detection on quantum data can be reduced to evaluating a smooth function of the average state's spectrum, f((C − μI)/T), without ever recovering eigenvectors, and that this soft thresholding recovers the hard-projector PCA score in the T→0 limit while being more robust to spectral noise near the cutoff. The dimension-independent calibration sample complexity for the residual Q score follows from the fact that the retained-mass curve R(μ) = Tr(f((C − μI)/T) C) is a one-dimensional monotone function of μ, so a standard concentration inequality (Dvoretzky–Kiefer–Wolfowitz) gives the O(η⁻² log δ⁻¹) bound regardless of d.

Load-bearing premise

The dimension-independent calibration sample complexity assumes that the Hamiltonian simulation primitive for the average state C—whether implemented via block-encoding or density matrix exponentiation—does not itself introduce costs that scale with the data dimension d. If the simulation of e^{itC/T} scales with d or with simulation time, the end-to-end complexity is not dimension-independent.

Editorial extensions

If this is right

  • Quantum-native systems with graded spectra—where no discrete rank cutoff is well-defined—can be monitored for anomalies using a single retained-mass target α and a resolution parameter T, avoiding the ill-posed rank selection of hard PCA.
  • The soft thresholding approach could be extended to multi-class quantum state discrimination, with separate calibrated spectral supports learned for each class.
  • The dimension-independent sample complexity for Q-score calibration (if the Hamiltonian simulation primitive is efficient) could make QSPADE practical for high-dimensional quantum systems where eigenvector recovery is infeasible.
  • The qumode implementation's ability to sweep all thresholds from a single set of measurement records could enable real-time spectral monitoring of quantum devices without recomputation for different retained-mass targets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The dimension-independent sample complexity claim depends on the cost of implementing e^{-ip̂⊗C/T₂} or e^{itωC/T}. If this simulation cost scales with d or with simulation time, the end-to-end complexity is not dimension-independent. The paper acknowledges this access model but does not fold simulation cost into the bound.
  • The soft Q statistic's graded penalty on boundary modes partially absorbs the function of Hotelling's T² near the cutoff, which could reduce the practical need for the more expensive T² calibration in settings where anomalies are expected near the spectral boundary rather than deep within the retained subspace.
  • The temperature parameter T introduces a resolution–robustness trade-off: smaller T sharpens the detector but may require more measurement shots to resolve the compressed acceptance probabilities, as shown in the finite-shot AUC analysis. This suggests an optimal T exists that balances spectral resolution against shot noise, which the paper explores empirically but does not derive analytically.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper proposes Quantum Spectral Anomaly Detection (QSPADE), a measurement-based framework for computing PCA-like anomaly scores directly from the spectrum of the average state C = ρ̄ of a normal quantum dataset. The core idea is to replace the hard rank selection of classical PCA with a smooth, temperature-controlled spectral threshold M^f_{μ,T} = f((C − μI)/T), where f is a monotone smooth response function. The detector recovers the hard-projector PCA score in the T→0 limit. The paper provides two quantum implementations: a qumode circuit using position-quadrature measurements and a qubit circuit using Hadamard tests with Fourier representations of f. Calibration of the residual Q statistic is shown to require O(η⁻² log δ⁻¹) measurement samples via a DKW inequality. Numerical experiments demonstrate kernel-PCA-like behavior on encoded classical data and detection of a transverse-field Ising phase transition without predefined order parameters. The mathematical core (Proposition III.2, Theorem IV.2, Propositions IV.3 and V.2) is proved in appendices using standard techniques.

Significance. The paper addresses a well-motivated problem: computing anomaly scores for quantum data without full eigenvector recovery or Gram matrix construction. The variational derivation of the soft spectral detector (Theorem IV.2) from a strictly convex optimization problem is clean and self-contained, and the sharp-limit correspondence with classical PCA Q and T² statistics (Proposition V.2) is a concrete, falsifiable claim. The qumode implementation that reuses a single set of position samples for all calibration targets via post-processing is an elegant design. The TFIM experiment demonstrating phase-transition detection without order parameters is a compelling proof-of-concept for quantum-native monitoring. The dimension-independent DKW calibration bound for the residual Q score is a notable efficiency claim, though its scope requires clarification (see Major Comments).

major comments (3)
  1. Section VI, paragraph beginning 'Throughout this section, we use C = ρ̄ as a Hamiltonian generator': The headline claim of dimension-independent sample complexity O(η⁻² log δ⁻¹) for calibrating the residual Q score (Eq. VI.10, repeated in Table I and the abstract) counts only measurement shots for the DKW bound. Each shot requires one execution of U_C = e^{-i p̂⊗C/T₂} (qumode) or e^{it_ω C/T} (qubit), and the cost of implementing this primitive is not folded into the bound. For the qumode route via density matrix exponentiation (DME), each shot consumes O(t²/ε) copies of C with t ~ 1/T₂, introducing a T-dependent factor. For the qubit route, Eq. VI.24 gives O(A_f² η⁻² log δ⁻¹) samples, but A_f (the block-encoding norm of the Fourier representation of f) is not bounded as a function of T; as T→0 the interval J = [-μ_max/T, (1-μ_min)/T] grows, and the number of Fourier terms needed for ε_f
  2. Section VI.C, Eq. VI.24 and surrounding text: The paper states that the qubit route sample count is O(A_f² η⁻² log δ⁻¹) and notes that this is 'multiplied by the average or worst-case cost of simulating e^{it_ω C/T} over the sampled frequencies.' However, no bound on A_f is provided for any specific response function f (e.g., Fermi-Dirac or Gaussian from Appendix E). Since the central efficiency claim distinguishes the qumode and qubit routes by their sample complexities, at least a rough characterization of how A_f scales with T (and potentially d) for the recommended response functions would substantially strengthen the contribution. Without this, the qubit route's complexity is incompletely characterized.
  3. Section VI.B, Eq. VI.14: The T² calibration sample complexity is stated as Õ(Ldη⁻² log δ⁻¹) based on Bernstein's inequality applied to the maximally mixed probe τ_mm = I/d. This bound requires preparing τ_mm and implementing the qumode circuit with τ_mm as input. The paper should clarify whether the preparation of τ_mm (or the ability to implement e^{-i p̂⊗(I/d)/T₂}) is assumed as a primitive, and if so, whether its cost is dimension-dependent. This is relevant to the claim that QSPADE 'breaks the naive Õ(d²) worst-case scaling.'
minor comments (6)
  1. Table I caption: The T² sample complexity is listed as 'eO(Ldη⁻² log δ⁻¹)' but Eq. VI.14 gives Õ((Ld/η² + d/η) log δ⁻¹). The table should match the equation or explain the simplification.
  2. Section VII.B, Figure 2: The y-axis label 'eigenvalue λ_j' uses a log scale but the axis values (10⁻⁵ to 10⁻¹) are not explicitly labeled as log. A minor axis-label clarification would help.
  3. Section VII.B: The perturbation amplitude 0.05 is mentioned in Appendix H.1 but not in the main text. Including it in the main-text description of the experimental setup would improve reproducibility.
  4. Eq. VI.10: The DKW bound gives S ≥ (1/2η²) log(2/δ), but the text states O(η⁻² log δ⁻¹). This is correct up to constants but the factor of 2 inside the log is dropped without comment.
  5. Section V.B, Eq. V.8: The rank-calibration condition Tr(M^f_{μ_ℓ,T}) = ℓ places integer-valued constraints on the trace of a soft detector. For finite T, this may not have an exact solution; the paper should briefly note that approximate satisfaction suffices.
  6. References [12] and [13] are both dated 2026 and appear to be companion works by overlapping author sets. The relationship to these prior works (particularly [13], which shares three co-authors) should be made explicit to clarify the novelty boundary.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies that the headline dimension-independent sample complexity bound for Q-residual calibration counts only measurement shots and does not fold in the per-shot cost of the Hamiltonian simulation primitive. We agree this needs to be made explicit in the manuscript and will revise accordingly. We also agree that bounding A_f as a function of T for specific response functions would strengthen the qubit-route characterization, and we will add this analysis. On the T^2 calibration point regarding preparation of the maximally mixed probe, we will clarify the assumptions.

read point-by-point responses

  1. Referee: Major Comment 1 (Section VI): The headline O(eta^-2 log delta^-1) bound counts only measurement shots, not the per-shot cost of implementing U_C. For DME, each shot consumes O(t^2/epsilon) copies with t ~ 1/T_2, introducing a T-dependent factor. For the qubit route, A_f is not bounded as a function of T; as T->0 the interval J grows and the number of Fourier terms grows.

    Authors: The referee is correct that the O(eta^-2 log delta^-1) bound counts only measurement shots for the DKW inequality and does not include the per-shot cost of implementing the Hamiltonian simulation primitive U_C. We agree this needs to be stated more explicitly in the manuscript to avoid giving a misleading impression of the total computational cost. We will revise the manuscript to clearly separate the measurement-shot complexity (which is genuinely dimension-independent) from the per-shot simulation cost, and to note that the total cost is the product of these two factors. For the qumode route via DME, we will add the explicit statement that each shot consumes O(t^2/epsilon) copies of C with t ~ 1/T_2, so the total copy complexity acquires a T-dependent factor. For the qubit route, we will note that the per-shot cost depends on simulating e^{i t_omega C/T} and that A_f grows as T decreases. We emphasize that the dimension-independence claim is specifically about the number of measurement shots needed for calibration, which is the novel efficiency aspect of the DKW-based approach; the per-shot simulation cost is a separate, standard Hamiltonian simulation cost that applies to any quantum algorithm using C as a Hamiltonian generator. We do not claim that the total end-to-end cost is dimension-independent, and we will make this distinction sharper in the revised text. revision: yes

  2. Referee: Major Comment 2 (Section VI.C, Eq. VI.24): No bound on A_f is provided for any specific response function f (e.g., Fermi-Dirac or Gaussian). A rough characterization of how A_f scales with T (and potentially d) for recommended response functions would substantially strengthen the contribution. Without this, the qubit route's complexity is incompletely characterized.

    Authors: The referee is right that without any bound on A_f, the qubit route's complexity is incompletely characterized. We will add an analysis of A_f for the Fermi-Dirac and Gaussian responses in the revised manuscript. Concretely, for the Fermi-Dirac response f_FD(x) = 1/(1+e^{-x}), a Fourier series representation on the interval J = [-mu_max/T, (1-mu_min)/T] has coefficients that decay geometrically, so the number of Fourier terms needed for approximation error epsilon_f is O(log(1/epsilon_f)), and A_f scales as O(|J|) = O(1/T) in the worst case. For the Gaussian response, the Fourier transform decays super-exponentially, so the number of terms is O(sqrt(log(1/epsilon_f))), but A_f still scales with the interval width |J| ~ 1/T. In both cases, A_f^2 contributes a factor of O(1/T^2) to the qubit-route sample complexity, giving a total qubit-route complexity of O(T^{-2} eta^{-2} log delta^{-1}) for the measurement shots, multiplied by the per-shot simulation cost. This T-dependence is the price of the sharper spectral resolution, and it is consistent with the T-dependent factor in the qumode DME route. We will include this characterization in a revised version of Section VI.C or an appendix, and we will update Table I to reflect the T-dependence in the qubit route. revision: yes

  3. Referee: Major Comment 3 (Section VI.B, Eq. VI.14): The T^2 calibration bound O~(Ld eta^-2 log delta^-1) requires preparing tau_mm = I/d and implementing the qumode circuit with tau_mm as input. The paper should clarify whether preparation of tau_mm is assumed as a primitive and whether its cost is dimension-dependent, relevant to the claim of 'breaking the naive O~(d^2) worst-case scaling.'

    Authors: The referee raises a valid point about the preparation of the maximally mixed probe tau_mm = I/d. We will clarify in the revised manuscript that the T^2 calibration assumes access to tau_mm as a primitive. In practice, tau_mm can be prepared by generating a Haar-random pure state |phi> (so that E[|phi><phi|] = I/d) or by using a fixed circuit that produces the maximally mixed state, such as applying random Pauli rotations or using a quantum pseudorandom state generator. The cost of preparing tau_mm is indeed dimension-dependent in general: Haar-random state preparation requires O(d) gates in the worst case, though pseudorandom circuits can achieve this with polylog(d) depth under appropriate cryptographic assumptions. However, the claim of 'breaking the naive O~(d^2) worst-case scaling' refers specifically to the measurement-shot complexity for calibrating the L window thresholds, which is O~(Ld) rather than O~(d^2). The d factor here comes from the fact that the relevant binomial success probability under tau_mm is p ~ L/d, and Bernstein's inequality exploits the reduced variance p(1-p) ~ L/d. The preparation cost of tau_mm is a separate overhead that we will explicitly acknowledge. We note that for the default Q-residual statistic, no tau_mm preparation is needed at all, and the measurement-shot complexity is genuinely dimension-independent. The T^2 statistic is an optional add-on for leverage monitoring, and its O~(Ld) shot complexity already represents a quadratic improvement over the naive O~(d^2) scaling for L << d. revision: partial

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central derivation (Theorem IV.2) is self-contained, and self-citations are to independent prior work, not load-bearing circular definitions.

full rationale

The paper's central theoretical result—Theorem IV.2, which derives M^f_{μ,T} = f((C−μI)/T) as the unique optimizer of the variational problem (IV.2)—is proven in Appendix B via standard convex optimization (Fréchet first-order conditions and strict convexity of the generator ϕ_f). No step in this derivation assumes the conclusion. The sharp-limit recovery of hard PCA (Proposition IV.3) follows from the boundary conditions of f, and the classical correspondence (Proposition V.2) follows by substitution. The calibration step (Eq. V.2, R^f_{C,T}(μ_α) = α) is a standard root-finding calibration, not a fitted-then-predicted quantity. The sample-complexity bounds (Eqs. VI.10, VI.24) are derived from external concentration inequalities (DKW, Hoeffding, Bernstein) applied to the measurement statistics, not from self-cited premises. The paper does cite its own prior work [12, 13, 16, 18] for context and for the Fermi–Dirac response example, but these citations are not load-bearing for the core derivation: Theorem IV.2 is self-contained, and the cited works provide motivation and specific response-function examples rather than premises that would make the result circular. The dimension-independent sample complexity O(η⁻² log δ⁻¹) is a genuine DKW bound on measurement shots; the concern that per-shot simulation costs are not folded into this bound is a correctness/completeness issue, not a circularity issue. No step reduces to its inputs by construction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities, particles, forces, or dimensions. The soft spectral detector M^f_{μ,T} is a new mathematical object (a measurement effect), not a postulated physical entity. The qumode and qubit circuits are constructions within standard quantum information theory. All free parameters (μ, T, α, γ) are standard calibration hyperparameters, not fitted constants presented as predictions.

free parameters (4)
  • μ (threshold) = calibrated empirically via R^f_{C,T}(μ) = α
    The spectral threshold μ is not derived from first principles; it is calibrated by matching the retained mass to a target α. This is a standard calibration step, not a fitted constant, but it is a parameter chosen from data.
  • T (temperature/resolution) = chosen manually (e.g., T=3×10⁻³ in §VII.B)
    The temperature T controls the smoothness of the spectral threshold. It is set by hand in the experiments based on the spectral scale of the tail. No principled automatic selection rule is given.
  • α (retained-mass target) = set by user (e.g., α=0.99 in §VII.B, α=0.88 in §VII.A)
    The retained-mass target α is the analogue of PCA rank selection. It is a user-chosen hyperparameter, not derived from the data.
  • γ (ridge parameter for T²) = set by user (γ≥0)
    The ridge parameter γ in the Hotelling T² score (Eq. V.12) stabilizes the inverse eigenvalue weighting. It is a standard regularization hyperparameter.
assumptions (4)
  • domain assumption Access to the average state C = ρ̄ as a Hamiltonian generator via a simulation oracle or block-encoding or density matrix exponentiation.
    Section VI, opening paragraph: 'Throughout this section, we use C = ρ̄ as a Hamiltonian generator. This access can be supplied either by a Hamiltonian-simulation oracle or block-encoding for C, or, in the sample-access setting, by density matrix exponentiation.' The cost of this primitive is excluded from the sample complexity bounds.
  • standard math The response function f is optimization-admissible (Definition IV.1): C¹, strictly increasing, limits 0 and 1, integrable quantile, centered.
    Definition IV.1 imposes regularity conditions on f. These are standard for the variational derivation but restrict the admissible class of response functions.
  • domain assumption The normal training states {ρ_i} are i.i.d. samples from a fixed normal distribution.
    The average state C = N⁻¹ Σ ρ_i is used as the population average. The DKW bound (Eq. VI.10) assumes i.i.d. samples from a fixed distribution.
  • domain assumption The Hilbert space dimension d is finite for the qumode and qubit implementations.
    The spectral decomposition C = Σ λ_j |u_j⟩⟨u_j| (Eq. III.4) and the maximally mixed probe τ_mm = I/d (Eq. VI.11) assume finite d. The T² calibration complexity Õ(Ld) (Eq. VI.14) explicitly depends on d.

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Cite this review

Pith. "Pith review of Quantum Spectral Anomaly Detection." pith.science (2026). https://pith.science/paper/3L6OYGAT

@misc{pith2026260705307,
  author       = {Pith},
  title        = {Pith review of: Quantum Spectral Anomaly Detection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3L6OYGAT}},
  note         = {Machine review of arXiv:2607.05307}
}
read the original abstract

A core task in quantum anomaly detection is to compute an anomaly score that quantifies how strongly a test quantum state deviates from a given quantum dataset assumed to be normal. Classically, principal component analysis (PCA) for centered data computes the anomaly score by evaluating the test sample relative to the subspace spanned by the selected leading eigenvectors. However, for quantum data that lack a standard centering, explicitly recovering principal eigenvectors, constructing full Gram matrices, or loading quantum-random-access-memory-style data can be more costly than estimating the anomaly score itself. To avoid these costs, we propose Quantum Spectral Anomaly Detection (QSPADE), which computes PCA-like anomaly scores directly from the spectrum of the average state of the normal dataset. By replacing hard PCA rank selection with a smooth, temperature-controlled spectral threshold, QSPADE makes near-threshold spectral components contribute partially to the anomaly score. This makes the score vary continuously rather than jump when a borderline component is included or excluded, and makes it less sensitive to noise or arbitrary hard cutoffs near the threshold. In the zero-temperature limit, QSPADE recovers the hard-projector PCA score. The proposed measurement-based quantum detector can be calibrated with a sample complexity independent of the data dimension. Numerical simulations show that QSPADE behaves like kernel-PCA on encoded classical data and detects changes across a transverse-field Ising transition without predefined order parameters. Consequently, QSPADE gives an efficient framework for both quantum-kernel anomaly detection on encoded classical data and the monitoring of quantum-native systems where diagnostic observables are unknown.

Figures

Figures reproduced from arXiv: 2607.05307 by the authors.

Figure 1
Figure 1. FIG. 1. Normal-calibrated percentile maps over [0 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Spectrum and soft occupations of the average nor [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Mean residual anomaly score for each normal mode [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: FIG. 5. QSPADE residual score of the TFIM ground state as [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Residual percentile on the moon dataset for three [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Detection ROC-AUC versus resolution [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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    Fermi–Dirac response The increasing Fermi–Dirac, or logistic, response is fFD(x) := 1 1 +e −x .(E.1) Its quantile and generator are f −1 FD(m) = log m 1−m ,(E.2) ϕFD(m) =mlogm+ (1−m) log(1−m).(E.3) Its derivative is the logistic probability density f ′ FD(x) = e−x (1 +e −x)2 =...

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    Gaussian response The Gaussian, or probit, response is the standard nor- mal cumulative distribution function fG(x) = Φ(x) := 1 2 1 + erf x√ 2 (E.8) where erf(·) denotes the error function. Its derivative is the standard normal density f ′ G(x) = 1√ 2π e−x2/2.(E.9) The quantil...

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    Experiments setup In the classical-data experiment of Section VII A, each classical pointx= (x 1, x2)∈[0,1] 2 is mapped to a fea- ture state|ϕ(x)⟩by one of the maps below; we write ˜xi = 2xi−1. All maps produce a proper pure state of unit Euclidean norm: the proportionality si...

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    Resolution–shot trade-off (quantum-native experiment) The QSPADE score is read off finite measurement shots: the acceptance⟨ψ|M f µα,T |ψ⟩is estimated fromm binary accept/reject outcomes, so the empirical residual carries binomial sampling noise of orderm −1/2. The data and ca...

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