Sample-optimal learning of quantum states using gentle measurements

Cristina Butucea; Henning Stein; Jan Johannes

REVIEW 3 major objections 5 minor 1 cited by

Sample-optimal learning of quantum states using gentle measurements

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that local α-gentle measurements force a sample complexity of order $1/(\epsilon^2 \alpha^2)$ for both quantum state certification and quantum tomography, and it exhibits a label-switching measurement that attains this…

desk verdict The paper's sample-optimal rates for gentle state learning are probably right, but a sign error inverts the Neyman-Pearson lemma and breaks the proof of Theorem 4 as written. read the letter →

arxiv 2505.24587 v3 pith:EL3EAQCE submitted 2025-05-30 quant-ph math.STstat.MLstat.TH

Cristina Butucea , Jan Johannes , Henning Stein This is my paper

classification quant-phmath.STstat.MLstat.TH MSC 81P6881P45 PACS 03.65.Ta03.67.-a

keywords gentlemeasurementslocallyquantumstatetomographycertificationdifferentialprivacydata-processinginequalityLabelSwitchsamplecomplexity

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

Gentle measurements are quantum measurements that yield an outcome while leaving the measured state nearly intact, at a prescribed trace distance α from the original. This paper asks what that gentleness costs in the number of copies needed to learn an unknown quantum state, and answers: the sample complexity for both quantum state certification and quantum tomography becomes $1/(\epsilon^2 \alpha^2)$, instead of the standard $1/\epsilon^2$. The cost is quadratic in the gentleness parameter: the gentler one insists on being, the more copies are needed. The paper proves a quantum data-processing inequality that captures this information loss, shows via a gentle Neyman-Pearson lemma that the bound is asymptotically tight for small α, and exhibits a measurement—the quantum Label Switch—that attains the rate.

What carries the argument

The load-bearing object is the quantum data-processing inequality (qDPI) of Theorem 3: for any α-gentle measurement on a state set containing pure states, $$$D^{{\mathrm{sym}}$}_{\mathrm{KL}}($P^{{\rho_1}}$_M \,\|\, $P^{{\rho_2}}$_M) \le \left(\frac{8\$\alpha$}{(1-2\$\alpha$)^2}\right)^2 \|\rho_1-\rho_2\|_{\mathrm{Tr}}^2.$$ It is proved by first strengthening the gentleness–differential privacy link (Lemma 7: α-gentleness implies $\delta=2\log((1+2\alpha)/(1-2\alpha))$ quantum differential privacy), then bounding outcome-probability ratios via eigenvalue gaps. On product measurements the KL divergence adds over copies, so the per-copy $\alpha^2$ factor becomes a sample-complexity penalty. The matching upper bound is the quantum Label Switch (qLS), which builds an α-gentle measurement from any two-outcome PVM by replacing each projector $P$ with a mixture $\sqrt{e^\delta/(e^\delta+1)}\,P + \sqrt{1/(e^\delta+1)}\,P^\perp$ for $\delta=4\,\mathrm{arctanh}(\alpha)$; the gentle quantum Neyman-Pearson test uses the same construction on the positive and negative parts of $\rho_0-\rho_1$ and shows the inequality is asymptotically sharp for small α.

What would settle it

Compute the exact differential-privacy parameter $\delta(\alpha)$ forced by $\alpha$-gentleness for a concrete two-outcome measurement such as the quantum Label Switch: Lemma 7 predicts $\delta$ scales linearly in $\alpha$, so a direct calculation yielding $\delta \asymp \alpha^2$ for small $\alpha$ would change the sample complexity to $1/(\epsilon^2\alpha^4)$ and refute the claimed rate.

Watch

Extended reading notes

Core claim

The paper's central claim is that local α-gentleness—requiring each single-copy measurement to leave its subsystem within trace distance α of the input—cuts the information that any measurement can extract. Its Theorem 3 bounds the symmetrized Kullback-Leibler divergence of the outcome distributions for two states by a constant times $\alpha^2$ times the squared trace distance, for all $\alpha<1/2$. From this, the paper derives matching lower and upper bounds: state certification and tomography of qubits to trace accuracy $\epsilon$ require $\Theta(1/(\epsilon^2 \alpha^2))$ copies, versus $\Theta(1/\epsilon^2)$ without the gentleness constraint. The upper bound is achieved by the quantum Label Switch (qLS), a measurement that takes any two-outcome projective measurement and makes it α-gentle by mixing the outcomes with probabilities tied to $\tanh(\alpha/4)$. A gentle quantum Neyman-Pearson lemma shows the qDPI is asymptotically sharp as $\alpha\to 0$, so the $\alpha^2$ penalty is not an artifact of the proof.

Load-bearing premise

The load-bearing premise is that a measurement which moves any state by at most $\alpha$ also changes its outcome probabilities by a privacy factor growing linearly in $\alpha$ (Lemma 7); if that factor actually grew like $\alpha^2$, the claimed sample cost would become $1/(\epsilon^2\alpha^4)$ rather than $1/(\epsilon^2\alpha^2)$.

Editorial extensions

If this is right

For qubit tomography and state certification under local α-gentleness, the sample complexity is $\Theta(1/(\epsilon^2\alpha^2))$, so any procedure that leaves each copy nearly intact must pay a full factor $1/\alpha^2$ over standard tomography.
The quantum Label Switch provides a general recipe: from any two-outcome projective measurement, one gets an α-gentle measurement whose information–destruction trade-off is optimal up to constants, and for qubits it attains the lower bound.
Because the qDPI is asymptotically sharp for small α, the $\alpha^2$ information loss is intrinsic to gentle measurements and cannot be removed by cleverer estimators.
For global gentleness, the situation splits: no coherent globally gentle measurement can certify all states for small enough α, and no product measurement can be globally gentle on product states at fixed α; only entangled measurements on product states can work.

Reading between the lines

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

If the $1/\alpha^2$ penalty extends to higher-dimensional systems, then any protocol that must leave its quantum state reusable—such as quantum backpropagation or quantum memories—will face the same additional sample cost; the paper proves the qubit case and suggests the rate depends on dimension.
Because the quantum Label Switch is literally randomized response applied to the measurement outcome, the classical local differential privacy literature becomes a design guide for gentle measurements; optimal LDP mechanisms may translate into optimal gentle POVMs.
A direct hardware test is feasible: implement the qLS circuit (CNOT with an ancilla and computational-basis measurement) on a single qubit, measure the post-measurement trace distance and the estimator's mean squared error, and check the predicted $1/(n\epsilon^2\alpha^2)$ scaling.
The qDPI could be used in reverse: since gentleness implies quantum differential privacy, existing lower bounds for quantum differentially private mechanisms may be reinterpreted as lower bounds for gentle learning, giving a unified route to sample-optimal gentle algorithms.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper has a real result: it defines local gentleness for product measurements, proves a quantum data-processing inequality that ties gentleness to sample complexity, and gives a matching construction (quantum Label Switch) achieving 1/(ε²α²) for qubit tomography and certification. The qDPI is genuinely reusable, and the improved gentleness-to-DP relation (α < 1/2 instead of α < 1/4) looks correct—the qLS itself has δ ≈ 4α, so the linear-in-α scaling in Lemma 7 is consistent and not a fragile assumption. The rate results are plausible and the achievability side genuinely closes the loop.

That said, there is a load-bearing flaw in the gentle Neyman-Pearson lemma. In Lemma 10, the test is defined to reject H0 on outcome 1, but the measurement (5) puts the larger weight e^δ/(e^δ+1) on P+, the positive part of ρ0−ρ1. Outcome 1 is therefore more likely under ρ0, which is H0—the decision rule is inverted. A direct trace gives the total error as 1 + (e^δ−1)/(e^δ+1)∥ρ0−ρ1∥, not 1 − ..., and in fact that sum exceeds 1 for small α. So the displayed identity is false, and the subsequent bound "TV ≥ c∥ρ0−ρ1∥" does not follow. The proof of Theorem 4 fails as written. The fix is straightforward: flip the decision rule, or use P+ as the positive part of ρ1−ρ0, and the intended 1 − c∥ρ0−ρ1∥ and the lower bound come out. This is a real error but a repairable one, and it does not undermine the main rate claim because the qLS construction independently attains the same rate.

There are also smaller proof hygiene issues: Lemma 11's statement has a stray "p²" and the qDPI proof's bound ∑λmin(Ey) ≤ Tr(I)/2 = 1 is written for qubits even though the theorem is stated for general d; it can be repaired via λmin(Ey) ≤ Tr(Ey)/d, giving ∑λmin ≤ 1. These are minor.

My take: this is a serious paper worth refereeing. The qDPI and qLS are the kind of tools people will reuse, and the rates are likely correct. The authors need to fix the Neyman-Pearson sign/orientation before the asymptotic-sharpness claims can stand. Send it to a careful referee, not the wastebin.

Referee Report

3 major / 5 minor

Summary. This paper studies quantum state certification and tomography under α-locally-gentle measurements. The authors define local gentleness, derive a relation between gentleness and quantum differential privacy, and prove a quantum data-processing inequality (Theorem 3) bounding the symmetrized KL divergence of the outcome distributions by a constant times α² times the squared trace distance. Using this inequality, they prove lower bounds n = Ω(1/(ε²α²)) for both state certification and tomography, and they construct the quantum Label Switch procedure, a gentle-ized version of basis measurements, together with estimators and tests attaining n = O(1/(ε²α²)). They also propose a gentle quantum Neyman-Pearson test and claim that it shows the qDPI is asymptotically sharp.

Significance. If the technical issues below are repaired, the paper gives a coherent and non-obvious picture: locally gentle measurements necessarily incur an extra 1/α² factor in sample complexity relative to non-gentle measurements, and the quantum Label Switch attains this rate. The main strengths are the explicit and essentially self-contained derivation of the gentleness-to-privacy bridge, the explicit implementable construction of the quantum Label Switch, and the fact that the rate claims are supported by both a lower bound and a matching attainability construction. The flaw in Lemma 10 and the proof of Theorem 4 does not appear to destroy the central 1/(ε²α²) rate, which relies on Theorem 3 and the qLS construction, but the sharpness/optimality claims need correction before they can be accepted.

major comments (3)

[Section 4, Eq. (5) and Lemma 10] The gentle Neyman-Pearson test as stated is inverted. Since P+ is the projector onto the positive part of ρ0−ρ1, outcome 1 is more likely under ρ0 than under ρ1, whereas the decision rule announces H1 on outcome 1 and H0 on outcome 0. A direct trace computation gives P0(Δ*α=1)+P1(Δ*α=0) = 1 + (e^δ−1)/(e^δ+1)∥ρ0−ρ1∥_Tr, not the displayed 1 − (e^δ−1)/(e^δ+1)∥ρ0−ρ1∥_Tr; the identity Tr(ρ0P+ + ρ1P−) = 1 − ∥ρ0−ρ1∥_Tr used in the proof is also incorrect, the correct value being 1 + ∥ρ0−ρ1∥_Tr. Consequently the proof of Theorem 4 as written yields only the trivial bound TV ≥ −c∥ρ0−ρ1∥. The intended result is restored by reversing the decision rule, i.e. announcing H1 on outcome 0, which gives total error 1 − 2α/(1+α²)∥ρ0−ρ1∥_Tr and hence the desired bound (6). Lemma 10 and the proof of Theorem 4 must be corrected accordingly.
[Theorem 3, statement and proof] The theorem is stated for M ∈ GM(α,S) where S is merely a set of quantum states containing the pure states, but the proof invokes Lemma 7, which assumes α-gentleness on all of S(C^d). The convex-mixture state ρλ = λρ1 + (1−λ)ρ2 used in the proof of Lemma 7 need not belong to S, so the cited lemma does not apply under the stated hypothesis. Since every application in Section 5 uses S = S(C^d), restricting the statement of Theorem 3 to S = S(C^d), or supplying a separate argument for general S, is a simple fix, but the theorem as stated is not proven.
[Theorem 3, proof after Eq. (3)] The displayed chain Σ_y λmin(Ey) ≤ Σ_y Tr(Ey)/2 ≤ Tr(I_d)/2 = 1 is not correct for general d: Tr(I_d)/2 = d/2, and the inequality λmin(Ey) ≤ Tr(Ey)/2 can fail when d > 2. The intended bound Σ_y λmin(Ey) ≤ 1 does follow from λmin(Ey) ≤ Tr(Ey)/d, so the conclusion of the theorem is unaffected, but the proof as written contains a false intermediate step and should be corrected.

minor comments (5)

[Lemma 12, proof] The displayed choice ε = min{1, (1−2α)^4/(6α√n)} does not produce the stated lower bound (1−2α)^4/(864nα²); substituting it into the preceding expression gives (1−2α)^8/(864nα²). Using ε = min{1, (1−2α)^2/(6α√n)} yields the stated bound. The Ω(1/(ε²α²)) rate is unaffected.
[Lemma 12, proof] The indicator Δ is defined on the event {∥ρhat−ρ0∥² ≥ ε/2}; it should be defined on {∥ρhat−ρ0∥_Tr ≥ ε/2}, and the same squared-norm typo appears one line earlier.
[Lemma 9 and Section 6] The definition δ = 4 arctanh(α) is not finite at α = 1. Since α = 1 is the non-gentle limiting case, the paper should state explicitly that this case is understood as a limit, or handle it separately.
[Lemma 6, proof] The notation for the coefficients ν̃_i is used ambiguously: it is written both as a probability Σ_j |ν_{i,j}| and as a coefficient √(Σ_j |ν_{i,j}|). Please clarify the notation so that the displayed ratio identity is unambiguous.
[References] The references [WB24a] and [WB24b] appear to describe the same paper with the same title and pages; please merge the duplicate entry.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 1/(ε²α²) rate follows from in-paper proofs of gentleness-to-privacy, the qDPI, and an explicit qLS construction.

full rationale

The derivation chain is self-contained. Lemma 7 proves the gentleness-to-quantum-differential-privacy relation inside the paper, and Theorem 3 derives the qDPI from that lemma together with Lemma 3 and eigenvalue bounds; it does not import the rate. The lower bounds in Lemmas 11 and 12 are obtained by applying the qDPI and Pinsker's inequality to the certification/testing problem, and the upper bounds in Lemmas 14 and 15 are obtained from the explicitly constructed quantum Label Switch, whose α-gentleness is proved in Lemma 9. The estimator's calibration factor (e^δ+1)/(e^δ-1) is computed directly from the measurement probabilities, not fitted to the target rate, so there is no fitted input renamed as a prediction. The gentle quantum Neyman-Pearson test in Lemma 10 and Theorem 4 is an explicit construction; optimality is argued by comparing its error to the minimal testing error, which is a standard inequality, not an assumed optimality. The only external citation for label switching, [Ste24], is used as motivation for the mechanism, while the quantum gentleness and statistical guarantees are proven in the paper, so it is not load-bearing. No self-referential definition ties α to 1/(ε²α²), and no uniqueness or optimality claim is imported solely from the authors' prior work.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard quantum measurement theory, classical statistical inequalities, and the classical local-DP lemma of DJW13, plus the paper's own Lemma 7. There are no free parameters fitted to data and no new physical entities introduced; the quantum Label Switch is a measurement gadget, not a new force or particle.

assumptions (5)

domain assumption Born rule and Kraus representation of measurements with post-measurement state update rho^{M->y} = M_y rho M_y^* / P(R=y).
Section 2.1; the entire framework of gentleness and the qDPI is built on this measurement model.
standard math Trace-norm properties, KL divergence, Pinsker's inequality, and KL additivity for product distributions.
Section 2.4; used in Theorem 3, Lemma 11, and Theorem 4.
standard math Lemma 4 of Duchi, Jordan, Wainwright (DJW13): for delta-DP mechanisms, D_sym_KL(P1||P2) <= sum_y |p1-p2|^2 / min(p1,p2).
Section 4, proof of Theorem 3; this classical privacy lemma is the bridge from eigenvalue ratios to the KL bound and is not reproduced in the paper.
standard math Schrodinger-HJW purification and contractivity of the trace norm under partial trace (to extend gentleness from pure to mixed states for positive operators).
Section 8.2, proof of Lemma 6; needed for qLS gentleness on all qubits.
standard math Naimark theorem, which lets the qLS POVM be implemented as a projective measurement on an enlarged Hilbert space.
Section 2.1 and Section 6.1; the physical implementation claim for qLS rests on this.

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Pith. "Pith review of Sample-optimal learning of quantum states using gentle measurements." pith.science (2026). https://pith.science/paper/EL3EAQCE

@misc{pith2026250524587,
  author       = {Pith},
  title        = {Pith review of: Sample-optimal learning of quantum states using gentle measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EL3EAQCE}},
  note         = {Machine review of arXiv:2505.24587}
}

abstract

Gentle measurements of quantum states do not entirely collapse the initial state. Instead, they provide a post-measurement state at a prescribed trace distance $\alpha$ from the initial state together with a random variable used for quantum learning of the initial state. We introduce here the class of $\alpha-$locally-gentle measurements ($\alpha-$LGM) on a finite dimensional quantum system which are product measurements on product states and prove a strong quantum Data-Processing Inequality (qDPI) on this class using an improved relation between gentleness and quantum differential privacy. We further show a gentle quantum Neyman-Pearson lemma which implies that our qDPI is asymptotically optimal (for small $\alpha$). This inequality is employed to show that the necessary number of quantum states for prescribed accuracy $\epsilon$ is of order $1/(\epsilon^2 \alpha^2)$ for both quantum tomography and quantum state certification. Finally, we propose an $\alpha-$LGM called quantum Label Switch that attains these bounds. It is a general implementable method to turn any two-outcome measurement into an $\alpha-$LGM.

Figures

Figures reproduced from arXiv: 2505.24587 by the authors.

**Figure 1.** Bloch sphere. Every qubit can be represented uniquely by a point in the ball of radius 1. 2.3 Distance measures of quantum states In order to quantify the closeness of two quantum states, we need to define distance measures on the space of quantum states. One of the most common distance measures for quantum states is the trace-norm distance. Definition 3 Let ρ1, ρ2 ∈ S(C d ) be two quantum states. The trace-norm be… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Optimal estimation of high-dimensional quantum states using locally gentle measurements
math.ST 2026-07 accept novelty 7.0 of 10

α-gentle tomography of rank-r qudits has minimax Frobenius rate Θ(rd²/(nα²)), with gentleness penalty scaling as ambient dimension d rather than parameter count rd.

Reference graph

Works this paper leans on

7 extracted references · 6 canonical work pages · cited by 1 Pith paper

[4]

Duchi, Michael I

[DJW13] John C. Duchi, Michael I. Jordan, and Martin J. Wainwright. Local privacy and statistical minimax rates. In2013 IEEE 54th Annual Symposium on Foundations of Computer Science—FOCS 2013, pages 429–438. IEEE Computer Soc., Los Alamitos, CA,

work page 2013
[22]

Explore google quantum ai.https://quantumai

[Teac] The Google Quantum AI Team. Explore google quantum ai.https://quantumai. google. Accessed: 2025-04-22. [Tsy09] Alexandre B. Tsybakov.Introduction to Nonparametric Estimation. Springer series in statistics. Springer,

work page 2025
[2016]

What is quantum computing?https://www.ibm

[SS] Josh Scheider and Ian Smalley. What is quantum computing?https://www.ibm. com/think/topics/quantum-computing. Accessed: 2025-04-22. [Ste24] Lukas Steinberger. Efficiency in local differential privacy.Ann. Statist., 52(5):2139– 2166,

work page 2025
[2020]

[BDKS20] Cristina Butucea, Amandine Dubois, Martin Kroll, and Adrien Saumard

©2020. [BDKS20] Cristina Butucea, Amandine Dubois, Martin Kroll, and Adrien Saumard. Local differential privacy: elbow effect in optimal density estimation and adaptation over Besov ellipsoids.Bernoulli, 26(3):1727–1764,

work page 2020
[2021]

Pauli measurements are not optimal for single-copy tomography.CoRR, abs/2502.18170,

[ADLY25] Jayadev Acharya, Abhilash Dharmavarapu, Yuhan Liu, and Nengkun Yu. Pauli measurements are not optimal for single-copy tomography.CoRR, abs/2502.18170,

arXiv
[2023]

Exponential separations between learning with and without quantum memory

[CCHL22] Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li. Exponential separations between learning with and without quantum memory. In2021 IEEE 62nd Annual Symposium on Foundations of Computer Science—FOCS 2021, pages 574–585. IEEE Computer Soc., Los Alamitos, CA, [2022]©2022. [CTD+80] Carlton M. Caves, Kip S. Thorne, Ronald W. P. Drever, Vernon ...

work page 2021
[2024]

Learning with privacy at scale

[Teaa] Apple Differential Privacy Team. Learning with privacy at scale. https:// machinelearning.apple.com/research/learning-with-privacy-at-scale. Ac- cessed: 2025-05-05. 40 [Teab] Azure Quantum Doc Team. The qubit in quantum computing. https://learn. microsoft.com/en-us/azure/quantum/concepts-the-qubit. Accessed: 2025-04-

work page 2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

Duchi, Michael I

[DJW13] John C. Duchi, Michael I. Jordan, and Martin J. Wainwright. Local privacy and statistical minimax rates. In2013 IEEE 54th Annual Symposium on Foundations of Computer Science—FOCS 2013, pages 429–438. IEEE Computer Soc., Los Alamitos, CA,

work page 2013

[2] [22]

Explore google quantum ai.https://quantumai

[Teac] The Google Quantum AI Team. Explore google quantum ai.https://quantumai. google. Accessed: 2025-04-22. [Tsy09] Alexandre B. Tsybakov.Introduction to Nonparametric Estimation. Springer series in statistics. Springer,

work page 2025

[3] [2016]

What is quantum computing?https://www.ibm

[SS] Josh Scheider and Ian Smalley. What is quantum computing?https://www.ibm. com/think/topics/quantum-computing. Accessed: 2025-04-22. [Ste24] Lukas Steinberger. Efficiency in local differential privacy.Ann. Statist., 52(5):2139– 2166,

work page 2025

[4] [2020]

[BDKS20] Cristina Butucea, Amandine Dubois, Martin Kroll, and Adrien Saumard

©2020. [BDKS20] Cristina Butucea, Amandine Dubois, Martin Kroll, and Adrien Saumard. Local differential privacy: elbow effect in optimal density estimation and adaptation over Besov ellipsoids.Bernoulli, 26(3):1727–1764,

work page 2020

[5] [2021]

Pauli measurements are not optimal for single-copy tomography.CoRR, abs/2502.18170,

[ADLY25] Jayadev Acharya, Abhilash Dharmavarapu, Yuhan Liu, and Nengkun Yu. Pauli measurements are not optimal for single-copy tomography.CoRR, abs/2502.18170,

arXiv

[6] [2023]

Exponential separations between learning with and without quantum memory

[CCHL22] Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li. Exponential separations between learning with and without quantum memory. In2021 IEEE 62nd Annual Symposium on Foundations of Computer Science—FOCS 2021, pages 574–585. IEEE Computer Soc., Los Alamitos, CA, [2022]©2022. [CTD+80] Carlton M. Caves, Kip S. Thorne, Ronald W. P. Drever, Vernon ...

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Learning with privacy at scale

[Teaa] Apple Differential Privacy Team. Learning with privacy at scale. https:// machinelearning.apple.com/research/learning-with-privacy-at-scale. Ac- cessed: 2025-05-05. 40 [Teab] Azure Quantum Doc Team. The qubit in quantum computing. https://learn. microsoft.com/en-us/azure/quantum/concepts-the-qubit. Accessed: 2025-04-

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