The Stabilizer Bootstrap of Quantum Machine Learning with up to 10000 qubits

Frederic T. Chong; Jinglei Cheng; Junyu Liu; Xulong Tang; Youtao Zhang; Yuqing Li

REVIEW 4 major objections 4 minor 22 references

The Stabilizer Bootstrap of Quantum Machine Learning with up to 10000 qubits

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Whether stabilizer bootstrap can pre-optimize a variational quantum circuit classically is set by the observable and entanglement structure, with constant success in some cases and exponential decay in others.

desk verdict The four exact sampling probabilities are a real, useful result, but the strong/weak taxonomy overreaches: it is built on an unvalidated proxy and fitted exponents, so the quantum-advantage boundary claims need a rewrite. read the letter →

arxiv 2412.11356 v1 pith:BDJAJE7D submitted 2024-12-16 quant-ph cs.AIcs.LG

Yuqing Li , Jinglei Cheng , Xulong Tang , Youtao Zhang , Frederic T. Chong , Junyu Liu This is my paper · ORCID

classification quant-phcs.AIcs.LG

keywords quantummachinelearningvariationalcircuitsstabilizerbootstrapCliffordGottesman-KnilltheoremclassicalsimulationadvantageBayesianoptimization

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 tries to establish that the stabilizer bootstrap—a classical pre-optimization of variational quantum machine-learning circuits by sampling Clifford angle assignments—has a sharply structured success rate. The central finding is that the probability of drawing a useful (nonzero) initial point from a single-layer ansatz is constant for some observable/entanglement pairings and exponentially small for the dual pairings; these are called strong and weak stabilizer enhancement. Because Clifford circuits are classically simulable, a constant success probability means the bootstrap can be run at thousands of qubits, while an exponentially small one means the classical handle disappears as the system grows. The paper supports this with induction proofs for the two extremal cases, with large-scale simulations on up to 10000 qubits and datasets up to 1000 samples for the intermediate regime, and with an observed exponential decline in search efficiency as dataset size grows. A sympathetic reader would care because this gives a concrete, testable criterion for when variational QML circuits can be practically classically initialized—and therefore for where quantum advantage might still be possible.

What carries the argument

The central object is the stabilizer bootstrap itself: a restricted variational circuit made entirely of Clifford operations by forcing every $R_y$ angle to lie in $\{0, \pi/2, \pi, -\pi/2\}$, so the state can be simulated classically through the Gottesman-Knill theorem. The bootstrap samples these discrete angles, evaluates a mean-squared-error loss from measurements of a chosen Pauli-string observable, and hands the best points to a Bayesian optimizer (random-forest surrogate with a greedy acquisition function) that refines the angles before any quantum execution. The workhorse quantity is the probability of nontrivial sampling—the chance that a random Clifford angle assignment yields measurement outcome $\pm 1$ instead of $0$. If that probability is high, the optimizer has signal to work with; if all samples are trivial, the acquisition function degenerates to random exploration (the paper shows the expected-improvement score becomes proportional to the predictive variance alone). The proofs proceed by mathematical induction on the qubit count, tracking how $1$-, $-1$-, and $0$-states of the $Z$-string observable evolve through the CNOT layer.

What would settle it

One concrete check: for a weak-enhancement configuration (e.g., linear CNOTs with a $Z$-string observable), enumerate or sample the Clifford angle assignments at $n=30$ and count the fraction with nonzero expectation; if it is not close to $1/2^{16}$, the exponential formula fails. A second check targets the proxy itself: train the same variational circuit after bootstrap initialization and after random initialization on several datasets of growing size; if a strong-enhancement configuration yields no persistent improvement in final loss or convergence speed, then the sampling probability is not the right measure of enhancement.

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Extended reading notes

Core claim

The paper's core claim is that the stabilizer bootstrap's effectiveness is governed by one number: the probability, over uniformly random choices of the discrete Clifford angles $\{0, \pi/2, \pi, -\pi/2\}$ for the $R_y$ layers, that a chosen Pauli-string observable has a nonzero expectation value on the prepared state. For single-layer ansatze with linear or reverse-linear CNOT entanglement, this probability is exactly $1/4$ for the strong pairings—linear CNOTs with an $X$-string, or reverse-linear CNOTs with a $Z$-string—and is $1/2^{\lceil n/2+1\rceil}$ for the weak pairings, namely linear CNOTs with a $Z$-string and reverse-linear CNOTs with an $X$-string. The strong pairings give constant probability independent of qubit number, so the bootstrap can classically find good initial points at any scale; the weak pairings lose this handle exponentially. For generic observables that mix $X$ and $Z$ factors (domain-wall strings), numerical experiments find the probability follows $p(r,n)=1/(4n^\nu)$, with the decay exponent $\nu$ interpolating between $0$ and $(n/2-1)\log 2/\log n$; most real cases lie between the two extremes. The paper also reports that as the number of training samples grows, the minimum loss achieved by sampling rises and the variance of sampled losses falls, indicating exponentially decreasing search efficiency with dataset size.

Load-bearing premise

The load-bearing premise is that the probability of getting a nonzero ($\pm 1$) measurement from a random Clifford-angle sample is a faithful measure of how much the stabilizer bootstrap improves the final trained circuit; this proxy is used throughout the paper but is never validated against end-to-end training outcomes such as final loss or accuracy.

Editorial extensions

If this is right

Variational circuits in the strong-enhancement regime can have their initial angles found classically with constant probability at essentially any qubit count, so the bootstrap is a reliable polynomial-time pre-optimizer for those ansatz/observable pairs.
Circuits in the weak-enhancement regime require exponentially many Clifford samples (order $2^{n/2}$) to find a nonzero starting point, so the bootstrap cannot pre-train them at large scale and quantum advantage is not undercut by this method.
For mixed $X/Z$ observables the decay is polynomial in $n$ with an exponent $\nu$ that can be measured; comparing $\nu$ across candidate ansatze gives a quantitative design criterion for choosing circuits that retain a quantum advantage.
Dataset size acts as an independent resource: increasing the number of training samples from 100 to 1000 lowers minimum loss quality and sampling variance roughly exponentially, so classical pre-optimization becomes harder as the learning task grows.
Because the whole pipeline (encoding, ansatz, measurement) is Clifford when restricted to these angles, the bootstrap connects fault-tolerant-era Clifford simulation techniques directly to near-term variational quantum machine learning.

Reading between the lines

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

Going beyond the paper's stated conclusions: the strong/weak split gives a practical classical-simulatability test—a configuration with constant bootstrap success is one where a classical observer can, with constant probability, find a provably good initial state, which makes claims of quantum advantage for that configuration harder to defend.
A natural testable extension is to replace the paper's proxy (probability of nonzero sampling) with direct end-to-end metrics such as final test accuracy or gradient variance, and check whether the strong/weak boundary still holds.
The observed $X/Z$ duality between linear and reverse-linear entanglement points to an underlying Clifford-group symmetry; proving the intermediate exponents $\nu(r)$ from stabilizer theory would turn the numerical interpolation into a phase diagram rather than a fit.
The dataset-size dependence suggests that data encoding is part of the classical-simulability boundary; testing different encoding maps (not just $\{0,1,2,3\}\to\{0,\pi,\pi/2,-\pi/2\}$) would reveal whether the exponential decline with dataset size is universal.

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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

Here's the short version. The exact probability calculations for single-layer Clifford ansätze (Theorems 1–4) are the real contribution. They are new—CAFQA doesn't count these—and they look correct (the induction is sloppy but the counting checks for n=1 and the symmetry arguments hold). If you work on variational initialization, those four probabilities are worth having. The 10000-qubit simulations are also a serious engineering effort.

The soft spot is exactly where the reader put it. The paper defines 'probability of nontrivial sampling' (getting ±1) as the key metric for stabilizer-bootstrap improvement and then uses the strong/weak split to talk about a boundary of quantum advantage. But it never connects that probability to end-to-end training success. A single ±1 expectation value shifts the loss away from zero; it doesn't guarantee a good gradient direction, a low loss, or a useful initialization. The dataset experiments report minimum loss and variance, not p(±1), with no error bars or repeated trials. So the chain from theorems to 'classical simulability boundary' is essentially a conjecture.

The intermediate exponents ν are also fitted, not derived. The paper is candid about this: Appendix C says the 1/4 upper bound is 'not a mathematical conclusion but derived from various experiments.' That honesty is to its credit, but it means the 'most situations fall in between' claim is an interpolation, not a proven classification. The proofs themselves contain typos and missing subscripts that make verification harder than it should be; they need a cleanup pass, not a rewrite.

I don't think this is a reject. The theorems are parameter-free and not circular, the extremes of the taxonomy are proven, and the paper explicitly flags what is empirical. The problem is the interpretive layer: claiming a 'trajectory for defining the boundaries of feasible quantum advantages' goes well beyond the evidence. A good referee would ask the authors to (1) validate p(±1) against actual training outcomes on a few small circuits, (2) add error bars, and (3) rephrase the quantum-advantage discussion as speculation. That's a substantial revision, but the mathematical core deserves to see the light of day.

Bottom line: send it to review. If I were working on initialization schemes, I'd cite the four probabilities. I'd bring it to a reading group to discuss what counts as evidence for classical simulability.

Referee Report

4 major / 4 minor

Summary. The paper proposes the stabilizer bootstrap, a method to pre-optimize variational quantum machine learning circuits by restricting rotation angles to Clifford values and simulating the resulting circuits classically via the Gottesman-Knill theorem. The central quantity is the probability p(±1) that a uniformly random Clifford angle assignment yields a nonzero (sign ±1) expectation value for a given observable and entanglement structure. The authors prove exact values of this probability for single-layer circuits with linear or reverse-linear CNOT entanglement and Pauli-X or Pauli-Z string observables (Table I and Appendix B), use these to define 'strong' versus 'weak' stabilizer enhancement, and fit an intermediate decay law p(r,n)=1/(4 n^ν) for mixed X/Z observables. They report high-performance simulations with up to 10000 qubits and datasets up to 1000 points, and from these they infer that the possibility of improvements from the stabilizer bootstrap depends on observable structure and dataset size, with implications for the boundary of classical simulability and quantum advantage.

Significance. If the exact counting results are correct, the paper provides a clean theoretical classification for a simple but nontrivial family of Clifford-constrained variational circuits, which is a useful step toward understanding when classical pre-optimization of QML circuits remains feasible. The demonstration of classical simulation at 10000 qubits using stabilizer techniques is technically impressive and illustrates the practical reach of Gottesman-Knill simulation. The strong/weak enhancement taxonomy, however, is only as valuable as the proxy on which it is built; the paper does not yet show that the nontrivial-sampling probability predicts actual improvement in downstream variational training. The fitted intermediate exponents and dataset-size scaling claims also lack statistical support. These gaps currently prevent the paper from fully supporting its broader claims about the boundary toward quantum advantages.

major comments (4)

[Appendix B, Theorem 1 proof] The induction proof contains an unjustified probability identity: the text states 'p(the first n qubits are in the 1-state) = p(the first n qubits are in the 1-state | the n-th qubit is |0⟩ or |1⟩)' and justifies it by saying the condition is necessary. Necessity does not equate unconditional and conditional probabilities; a factor of p(the n-th qubit is |0⟩ or |1⟩) is missing. The proof also mixes the induction index k with the summation index k and uses undefined bit-string notation. Since Theorem 1 is the basis for the constant probability 1/4 in Table I and for the strong-enhancement classification, this gap must be fixed before the central classification is rigorous.
[Appendix B, Theorem 2 proof] The induction proof for the exponential-decay case has multiple typographical and logical issues: the reference 'additional table ??' is left unresolved in both the odd and even parts; factors such as 1/8 · 1/2^{k+2} appear without a clear derivation of where the 1/8 comes from; and the bit indices in expressions like |00x_{i3}...x_{in+2}⟩ are inconsistent with the summation index. Because Theorem 2 supplies the exponential decay that defines weak stabilizer enhancement, the proof needs to be rewritten carefully with correct notation and complete case analysis before it can be considered verified.
[Section II, Eq. (1) and FIG. 6] The decay law p(r,n)=1/(4 n^ν) is introduced as a hypothesis and the exponent ν is then fitted to the very data that the formula is intended to characterize. No error bars, confidence intervals, or goodness-of-fit measures are reported, and Appendix C explicitly states that the maximum probability 1/4 is 'not a mathematical conclusion but derived from various experiments.' Thus the intermediate 'most situations fall in between' claim is an empirical extrapolation, not a derived result, and it cannot support a rigorous strong/weak dichotomy as presented.
[Section II and FIG. 7] The claim that 'as the dataset size increases, the search efficiency declines exponentially' is based on plots of minimum loss and variance with no error bars, no repeated trials, and no fitted decay function. More importantly, the paper never validates the central proxy: a nontrivial sample (expectation ±1) is asserted to be the key metric for sampling efficiency, but no experiment demonstrates that higher p(±1) correlates with lower loss, faster convergence, or better final accuracy in the dataset experiments. Without this validation, the strong/weak enhancement taxonomy may not reflect actual end-to-end QML performance.

minor comments (4)

[Section II, paragraph on general observables] The sentence 'The experimental results for the observables consisting of X operators and Z operators and Clifford stabilizer are shown in FIG.2' appears to reference the wrong figure; the relevant results are in FIG.4 and FIG.5, not the ansatz diagram in FIG.2.
[Section II, FIG. 6 caption] The caption contains a typo: 'exponent carve' should be 'exponent curve', and 'Equation II' should refer to a numbered equation rather than a section.
[Appendix B, Theorem 2 proof] The proof would benefit from a preliminary lemma that tracks the probability distribution over the first two qubits, since the current text refers to 'TABLE II' for the two-qubit case but leaves the table reference incomplete in the even case.
[Appendix A, GP/EI derivation] The statement that setting the noise parameter σ_n^2 to 0 yields μ(x)=0 is correct only when all observed values y are zero; this assumption should be stated explicitly in the derivation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are self-contained induction proofs over Clifford circuits, and the fitted exponent is presented as an empirical characterization rather than as a prediction.

full rationale

The paper's central theoretical results (Theorems 1-4 in Appendix B) are derived by mathematical induction from the explicit Clifford ansatz with Ry angles restricted to {0, pi/2, pi, -pi/2} and the observable being a Pauli-X or Pauli-Z string. These proofs do not invoke fitted parameters, empirical data, or prior work; they are parameter-free counting arguments. The probability of measuring 1 or -1 is computed directly from the stabilizer circuit structure, so the strong/weak classification for the four extreme cases is not circular. The intermediate-regime claim ('most situations fall in between') is supported by the fitted exponent nu in Eq. (II), but the paper explicitly describes this as an approximation of decay curves ('we try to use the following function to approximate the decay curves'), and the strong and weak extremes are proven independently. The CAFQA citation [8] is motivational background for the stabilizer bootstrap method, not a load-bearing premise for the new probability theorems; the paper also acknowledges that the maximal probability bound 1/4 is 'not a mathematical conclusion but derived from various experiments.' No step was found where the derivation reduces by construction to its own inputs or where a fitted parameter is relabeled as a prediction. The p(1) metric is a chosen proxy for sampling efficiency, but the paper does not define 'improvement' as p(1); it proves the metric's values and separately argues the importance of nontrivial sampling in Appendix A. Thus no circularity is established.

Assumptions & free parameters 2 free parameters · 4 assumptions · 3 invented entities

The central classification into strong/weak enhancement depends on the exact probabilities from the theorems (which are parameter-free) and on the assumed power-law form for intermediate cases (which introduces the fitted exponent ν). The proxy definition of 'nontrivial sampling' as a nonzero expectation value is a domain assumption that is not validated against downstream training performance.

free parameters (2)

critical exponent ν = varies with r and n; e.g., ν=0 for strong, ν=(n/2−1)log2/log n for weak extremes
The decay p(r,n)=1/(4 n^ν) is fitted to simulation data for intermediate observable mixtures; no first-principles derivation is given for this functional form.
dataset feature-to-angle mapping = features mapped to {0,1,2,3} -> angles {0,π,π/2,−π/2}
This choice ensures the encoding circuit is Clifford; other encodings could change the results.

assumptions (4)

standard math Gottesman-Knill theorem: Clifford circuits can be classically simulated in polynomial time
Used to justify simulating up to 10,000 qubits on a classical cluster.
domain assumption The measurement 'outcome' is defined as the sign of the Pauli expectation value, with 0 for states not in a ±1 eigenspace
This is how the probability of nontrivial sampling is defined; it is not a projective measurement but an expectation-value proxy.
domain assumption Uniform distribution over the four angles {0,π/2,π,−π/2} when computing sampling probability
The theorems count eigenstates under the assumption that each angle is equally likely.
ad hoc to paper The decay law p(r,n)=1/(4 n^ν) for intermediate observables is extrapolated from experiments
No proof is provided that all observable mixtures obey this power-law decay; it is a fit to simulation data.

invented entities (3)

Strong stabilizer enhancement
purpose: Label for configurations where the probability of nontrivial sampling stays constant (1/4) as qubits grow
A descriptive category for the already-derived probabilities; not a new physical entity.
Weak stabilizer enhancement
purpose: Label for configurations where the probability decays exponentially
Same, descriptive.
Critical exponent ν
purpose: Quantifies the decay rate as p=1/(4 n^ν) to interpolate between constant and exponential regimes
An invented quantity defined by fitting the assumed power-law model.

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Cite this review

Pith. "Pith review of The Stabilizer Bootstrap of Quantum Machine Learning with up to 10000 qubits." pith.science (2026). https://pith.science/paper/BDJAJE7D

@misc{pith2026241211356,
  author       = {Pith},
  title        = {Pith review of: The Stabilizer Bootstrap of Quantum Machine Learning with up to 10000 qubits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BDJAJE7D}},
  note         = {Machine review of arXiv:2412.11356}
}

read the original abstract

Quantum machine learning is considered one of the flagship applications of quantum computers, where variational quantum circuits could be the leading paradigm both in the near-term quantum devices and the early fault-tolerant quantum computers. However, it is not clear how to identify the regime of quantum advantages from these circuits, and there is no explicit theory to guide the practical design of variational ansatze to achieve better performance. We address these challenges with the stabilizer bootstrap, a method that uses stabilizer-based techniques to optimize quantum neural networks before their quantum execution, together with theoretical proofs and high-performance computing with 10000 qubits or random datasets up to 1000 data. We find that, in a general setup of variational ansatze, the possibility of improvements from the stabilizer bootstrap depends on the structure of the observables and the size of the datasets. The results reveal that configurations exhibit two distinct behaviors: some maintain a constant probability of circuit improvement, while others show an exponential decay in improvement probability as qubit numbers increase. These patterns are termed strong stabilizer enhancement and weak stabilizer enhancement, respectively, with most situations falling in between. Our work seamlessly bridges techniques from fault-tolerant quantum computing with applications of variational quantum algorithms. Not only does it offer practical insights for designing variational circuits tailored to large-scale machine learning challenges, but it also maps out a clear trajectory for defining the boundaries of feasible and practical quantum advantages.

Figures

Figures reproduced from arXiv: 2412.11356 by the authors.

**Figure 3.** FIG. 3: Result for 10000 qubits with 200 data. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 2.** FIG. 2: Clifford ans¨atze. For simplification, We refer to [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: FIG. 4. Considering the parity issue shown in condition [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 7.** Figure 7: FIG. 7: Search efficiency under different dataset sizes [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 6.** Figure 6: FIG. 6: Exponents for even and odd number of qubits. [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 8.** Figure 8: FIG. 8: Overview of QML [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: FIG. 9: Overview of Bayesian optimization. [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: FIG. 10: Proof details in Theorem 1 with Position A and Position B. [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: FIG. 11: Proof details in Theorem 2 with Position [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

**Figure 13.** Figure 13: FIG. 13: Probability of nontrivial sampling with multi-layer ans¨atze. [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

**Figure 12.** Figure 12: FIG. 12: Probability under all possible observables. [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 14.** Figure 14: FIG. 14: Results for datasets of varying sizes. [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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