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Statistical Symmetry Release for Equivariant Quantum Learning

Zeyu Chen

arXiv:2609.11470v1cs.LOquant-ph

TL;DR

Hard symmetry can erase label information, raising when finite data and measurement access justify relaxing the constraint. The paper links global signal detection to certified local release and validation, and reports that shared measurements required 6300 times fewer shots than the specified scalar estimator on the tested grids.

  • Problem

    Hard symmetry can erase label information, so the paper asks when finite data and measurement budgets justify relaxing the constraint and which available directions improve the chosen loss.

    Source: §1 · p. 1

  • Method

    The paper connects global symmetry-breaking signal tests to local loss-curvature certificates and independent validation, with shared Pauli measurements or scalar probes supporting direction selection.

    Source: §11 · p. 47

  • Results

    On the tested grids, the two specified Ising estimators differed by a factor of 6300 in the shots required to certify the leading release direction.

    Source: §11 · p. 47

  • Takeaways & Limitations

    Symmetry relaxation is treated as a model-selection decision grounded in detected task signal, certified local improvement, and validation of the trained model.

    Source: §11 · p. 47

    Establishing minimax complexity for an unknown composite symmetric null remains open.

    Source: §3.4 · p. 13

Abstract

from arXiv · show

Hard symmetry constraints reduce model complexity, but can also erase label information. Statistical symmetry release determines when finite data and quantum measurements justify relaxing such a constraint, which directions to open, and how far to move. We connect global signal detection to local, loss-dependent improvement. A two-copy twirl--swap gate estimates task information in the symmetry-breaking complement with a dimension-independent copy count under paired-state and group-unitary access; reweighting the same records resolves representation sectors. An exact duality distinguishes this Hilbert--Schmidt signal from the larger signal accessible to bounded-outcome readouts. Local improvement is governed by the release gradient and a loss-corrected double-commutator matrix. Simultaneous confidence bounds convert empirical direction selection into certified descent, using either shared Pauli measurements or scalar probes with state-independent truncation bounds. Gaussian testing lower bounds quantify the cost of searching over unknown directions in the calibrated local experiment. Independent validation controls adaptively generated models, and a fast squared-loss bound preserves the approximation--estimation rate of a nested release path. On an eight-qubit Ising model, shared measurements certify release with 6300 times fewer shots than the specified scalar estimator on the tested budget grids. Quotient quantum natural gradient then controls parameter redundancy during training. Together, these results turn symmetry relaxation into a statistically justified model-selection decision.

1 Introduction

Hard symmetry can make models simpler while erasing label information, motivating a data- and measurement-dependent decision about when and how to relax it. The paper distinguishes global detection of symmetry-breaking signal from local, loss-dependent improvement and places that decision alongside existing equivariant models and training methods.

1 Introduction

  • Hard invariant readouts cannot distinguish states with the same group twirl, so spin-flipped classes with opposite magnetization can remain indistinguishable despite label-relevant odd magnetization.Optimization within the invariant class leaves this information inaccessible.Source: §1 · p. 1
  • A global test asks whether breaking readouts contain label information, whereas a local test asks whether an available circuit perturbation reduces the chosen loss.A global signal may lie outside implementable tangent directions, and searching noisy local estimates can produce apparent improvement without true improvement.Source: §1 · p. 1
  • A two-copy twirl–swap gate estimates squared task signal in the symmetry-breaking complement with a Hilbert-dimension-independent copy count under the specified access model.Reweighting the same records resolves representation sectors, while a duality characterizes the strongest bounded-outcome signal as a trace-norm distance to the commutant.Source: §1 · pp. 1–2
  • Local release uses the descent gradient and loss-corrected curvature to certify a selected direction, with shared Pauli records requiring 104 shots versus 6.3 × 10^7 for the specified scalar estimator on the tested Ising instance.The paper also uses Gaussian testing bounds for unknown-direction search, independent validation for adaptively generated models, and a fast squared-loss bound for nested release paths.Source: §1 · p. 2

1.1 Relation to existing quantum equivariant learning

  • Prior equivariant architectures and horizontal quantum gates specify compatible or relaxed model capacity, while this work tests whether added capacity is supported by task evidence and available measurements.Symmetry discovery supplies candidate groups, and quotient quantum natural gradient handles state-preserving parameter redundancy after release.Source: §1.1 · p. 2, §1 · p. 2

2 Problem Setting: Symmetry as a Model-Selection Variable

The paper treats symmetry as a model-selection variable whose candidate structures, measurement access, and validation procedure determine the scope of each guarantee. It formalizes hard and soft release, explains what twirling hides, and balances approximation gains against the costs of opening directions.

2 Problem Setting: Symmetry as a Model-Selection Variable

  • A candidate structure specifies a hard backbone symmetry, an activated breaking subspace, and soft amplitudes, while guarantees depend on the candidate set and measurement interface.The predictor class and risk are defined for each structure, and a trained output can have risk above the class optimum.Source: §2 · p. 2, §2.1 · p. 3

2.1 Data, group actions, and candidate structures

  • Hard H-symmetry restricts linear readouts to the Hermitian commutant, while the statistical results require each candidate to yield an auditable finite- or effectively finite-dimensional class.Opening an H-breaking direction generally reduces the exact symmetry; a closed gate removes that direction, whereas a positive soft gate allows amplitude.Source: §2.1 · p. 3
  • Soft gates use s_j(α_j)=1/(1+α_j), with α_j=∞ retaining a direction and α_j=0 giving full local release.With unrestricted ungated coordinates, a finite nonzero gate is only a reparameterization; it changes the feasible class under a fixed-radius constraint.Source: §2.1 · p. 3

2.2 Sample splitting, measurement access, and adaptive candidate families

  • Three independent streams separate parameter fitting and release searches from final structural-risk comparison, preserving validation independence after conditioning on the prevalidation data.The candidate family may depend on fitting and release streams, but not on the validation stream.Source: §2.2 · pp. 3–4
  • Paired-copy, matrix-element, and scalar-probe modes estimate release evidence, and each requires simultaneous confidence coverage before its output enters the release criterion.Empirical eigendirections can reuse a simultaneously estimated matrix, but adaptively proposing additional observables requires a fresh batch or an enlarged uniform bound.Source: §2.2 · p. 4

2.3 Twirling blind spots and the release complement

  • Hard H-invariant readouts depend only on the twirled state, and squared-loss prediction incurs irreducible bias for label components orthogonal to the twirled feature.Among predictors measurable from that feature, the optimum is the conditional expectation; a restricted linear class may incur additional approximation error.Source: §2.3 · p. 5
  • In the spin-flip example, the class twirls coincide and every hard invariant measurement is at chance, while normalized magnetization distinguishes the classes with gap 2m.The released-sector classification test uses M copies and has an error bound from thresholding its empirical mean.Source: §2.3 · p. 6

2.4 The two costs of retaining or releasing symmetry

  • For squared loss, any broken group element forces approximation bias relative to the equivariant predictor projection.The bound relates the projection error to the discrepancy between the target predictor and its transform under the broken element.Source: §2.4 · p. 6

3 Quantum-Native Gate before Local Release

The global gate tests whether labels carry information in the symmetry-breaking complement before a release direction is chosen. Twirl–swap overlap estimation removes the symmetric component, while character reweighting resolves the remaining signal by representation sector.

3.1 Task-conditional breaking mass

  • Task-conditional breaking mass measures label-separating symmetry breaking, unlike the single-state asymmetry mass; a sign-equivariant task can retain odd-sector signal despite passing a joint symmetry test.The joint input–label test and task breaking mass concern different quantities.Source: §3.1 · p. 8
  • For bounded-outcome readouts, the breaking signal follows a different norm duality: involutive actions attain the trace-norm signal, and general bounds depend on the breaking rank.Retention is certified relative to the declared readout norm and task-relevance scale.Source: §3.1 · p. 9

3.2 Two-copy twirl–swap gate

  • With reused finite data, the shot interval targets empirical class means; transferring it to population means requires adding a data-sampling error bound.Fresh independent labeled examples instead give the product-state expectations for population class-conditional states.Source: §3.2 · p. 9
  • Both twirl–swap gates use M = O(r^-4 log(1/δ)) copy pairs, independent of Hilbert dimension, breaking dimension, and dictionary size, conditional on the stated access model.The access model assumes independent state copies, implemented Haar-random group elements, and SWAP measurements.Source: §3.2 · pp. 9–10

3.3 Shared-sample sector spectroscopy

  • One destructive-measurement sample set resolves all active representation-sector masses, with accuracy r² and only logarithmic dependence on the number of active sectors.Sector intervals can prioritize the local dictionary, but exclusion requires an upper confidence bound below the relevance scale.Source: §3.3 · pp. 11–12

3.4 Access-model boundary and limits of the gate

  • The paired-copy gate is favorable in high-dimensional, shot-limited settings, but direction-resolved measurements can sharpen with more shots, making screening access-dependent rather than mandatory.A structured Pauli-spike alternative detects breaking mass r² with copy count of order r^-2 up to logarithms.Source: §3.4 · p. 12
  • The minimax copy complexity for an unknown composite symmetric null remains open, and the general spectrum-testing lower bound does not settle the restricted Pauli-spike family.The stated O(r^-2) upper bound for that family does not establish its minimax complexity.Source: §3.4 · p. 13

4 Multi-Direction Release: Curvature Matrices and Orthogonalization

The release curvature matrix compares combinations of breaking generators, while nonlinear supervised losses require loss-dependent corrections and local gain can also include first-order release gradients. Gram and Fisher geometries normalize the dictionary and distinguish operator scale from local distinguishability.

4 Multi-Direction Release: Curvature Matrices and Orthogonalization

  • A quadratic form over the full generator dictionary can identify favorable combinations even when no individual diagonal curvature is positive.Its spectrum identifies favorable combinations, and Gram normalization makes their relative scale explicit.Source: §4 · p. 13

4.1 Symmetrized double-commutator curvature matrix

  • For nonlinear supervised losses, the release curvature combines loss-weighted readout curvatures with a second-derivative correction, both needed to compare directions under the actual loss.When first-order readout responses vanish, the correction disappears and the curvature reduces to the double-commutator form.Source: §4.1 · pp. 13–14
  • At full-block first-order stationarity, a positive eigenvalue of the release curvature matrix is necessary and sufficient for a strictly negative quadratic loss variation; zero modes need higher-order analysis.For a minimization objective, a positive quadratic form along a generator combination gives a second-order local decrease.Source: §4.1 · pp. 13–14

4.2 Local release gain with first-order terms

  • With first-order release gradients and declared activation and structural costs, a confidence-certified trust-region choice can yield a local decrease in penalized validation risk.The guarantee accounts for simultaneous estimation error and a bound on the third-order remainder.Source: §4.2 · p. 15, Assume also that the simultaneous… · p. 15

4.3 Hilbert–Schmidt and Fisher orthogonalization

  • Hilbert–Schmidt Gram normalization sets an offline operator scale, whereas the Fisher Gram measures local physical distinguishability; Fisher-null directions cannot be certified from a point estimate alone.Such directions require another probe state, validation point, or quotient treatment.Source: §4.3 · p. 16

5 Finite-Shot Certified Release

Finite-shot release is certified by adjusting curvature for declared costs and statistical uncertainty, then checking whether a selected direction yields descent. Shared measurements or scalar probes provide executable curvature estimates, while a local quartic model suggests release amplitude.

5.1 Penalized spectral release

  • A positive eigenvalue of K − SM − 2GN gives a direction of negative second-order change in the penalized local model; a negative-semidefinite matrix rules out second-order release descent.This criterion does not resolve higher-order descent on a zero mode.Source: §5.1 · p. 17

5.2 Simultaneous spectral confidence and subspace recovery

  • Under a simultaneous curvature confidence event, an empirical eigenvalue above the uncertainty radius certifies a positive population eigenvalue, while an empirical top eigenvalue below the negative radius rules out quadratic gain.Eigenvalues inside the confidence margin remain statistically ambiguous and should not be forced open.Source: §5.2 · p. 17
  • A selected empirical direction yields certified population descent when its confidence-adjusted gain is positive and the proposed step satisfies the smoothness condition.The leading empirical eigenvalue exceeding the confidence radius is sufficient when both declared penalties are positive semidefinite.Source: §5.2 · p. 17

5.3 Finite-shot executable estimation: shadow matrices and scalar probes

  • Simultaneous curvature estimation uses shared shadow records or bounded scalar snapshots, with entrywise uncertainty combined into an operator-norm bound valid after direction selection.For a fixed empirical supervised risk, snapshots sampled uniformly with replacement yield a bound conditional on the selected examples.Source: §5.3 · pp. 19–20
  • Low-weight local release dictionaries can have polynomial measurement overhead when grouped Pauli coefficient norms and candidate-pair counts are polynomial; high-weight commutators may instead favor scalar probes.Scalar central differences measure the original loss observable, with a step schedule of δβ ≍ M^-1/8 described for the bias–variance tradeoff.Source: §5.3 · pp. 20–21

5.4 How far to release

  • The quartic surrogate suggests a continuous onset of release amplitude, but an approximate quartic coefficient alone does not certify a finite step.A smoothness certificate or independent validation is still required.Source: §5.4 · p. 22

6 Local Statistical Decision Boundary for Releasing Symmetry

The paper models local release detection as Gaussian testing, deriving signal thresholds for unknown breaking directions and for searching a direction dictionary. These boundaries apply to quantum release only when the statistic and noise scale are calibrated to the experiment.

6 Local Statistical Decision Boundary for Releasing Symmetry

  • In the calibrated k-dimensional Gaussian release experiment, the detection threshold depends on the squared mean signal and noise scale; below the stated boundary, no procedure uniformly achieves both low false-release probability and high power.The threshold concerns unknown breaking directions, not a specific quantum measurement protocol.Source: §6 · p. 22
  • At fixed confidence, the squared mean-signal threshold in the Gaussian sequence experiment is of order τ²k.A curvature-search rule inherits this boundary only if its statistic and noise scale match the calibrated model.Source: §6 · p. 23

The threshold test

  • Searching over p candidate directions has an unavoidable signal scale involving τ√(log p), and below that scale false-release and worst-case missed-release probabilities sum to nearly one.The upper test controls type-I error without requiring independence among correlated dictionary statistics.Source: The threshold test · pp. 23–24
  • For correlated dictionaries, the search lower bound applies with the size of an orthonormal subdictionary, while the full-dictionary max test supplies an upper bound; a matching general metric-entropy characterization remains open.An unresolved branch may remain closed at the chosen scale or be considered as a separately validated soft-release candidate.Source: The threshold test · p. 24

7 Structural Risk Theory for Release Paths

Structural risk analysis balances the approximation gains of opening symmetry-breaking directions against estimation, measurement, optimization, and discovery costs. The section develops complexity and validation bounds for hard, soft, and adaptively selected release paths.

7.1 Projection bias from excessive symmetry

  • Along a genuinely nested release path, enlarging the allowed structure cannot increase approximation error, so estimation costs determine whether the bias reduction is worthwhile.The stated nesting for gated linear classes assumes a common coefficient-radius constraint.Source: §7.1 · p. 24

7.2 Effective commutant complexity

  • For a fixed class with bounded Hilbert–Schmidt coefficient radius, the data-dependent effective-commutant bound uses feature trace, which is bounded by input purity for an orthonormal operator basis.Capacity comparisons require the same readout normalization on both sides.Source: §7.2 · p. 25

7.3 Soft release effective dimension

  • Soft-release effective dimension weights breaking-coordinate covariance by gate strength, and each coordinate's feature variance and local-ridge shot cost scale with the square of its gate amplitude.A data-chosen gate still requires independent validation or uniform control for a population-risk bound.Source: §7.3 · pp. 25–26

7.4 Oracle inequality for candidate release paths

  • A finite-family structural-risk oracle inequality compares trained candidates using empirical risk plus nonnegative complexity, shot/release, and discovery costs.Its component-wise risk decomposition depends on corresponding bounds for the candidate trainers.Source: §7.4 · pp. 26–27

7.5 Post-selection validity: independent validation oracle

  • Independent validation controls selection among adaptively generated finite candidate families, and the squared-loss result preserves a nested-path approximation–estimation rate when shot and prevalidation costs are small enough.The fast validation rate uses the squared-loss variance relation; bounded loss alone gives a slower validation term.Source: §7.5 · pp. 27–28

7.6 Dynamical Lie algebra growth and barren-plateau margins

  • Under the stated curvature-concentration model, exponentially flat release branches are not certified with high probability when shot resources and the concentration scale grow only polynomially.The conclusion is conditional on the specified probing distribution and operator-norm concentration assumption.Source: §7.6 · p. 28, Suppose the empirical curvature… · p. 29

8 Quotient QNG after Symmetry Release

The section makes quantum natural-gradient training release-aware by accounting for parameter redundancies and separating horizontal updates from vertical drift. It connects physical release directions to a joint certificate for selection, descent, validation, and projector leakage.

8.1 Release-dependent vertical and horizontal spaces

  • On a regular stratum, the quotient removes state-preserving parameter redundancies under Γ-basic faithfulness, and its horizontal projector must be recomputed after release or pruning.Predictor invariance alone does not imply a null direction of the state quantum Fisher matrix.Source: §8.1 · p. 29

8.2 Gauge-projected Tikhonov QNG

  • An exact horizontal projector eliminates vertical shot-noise updates, whereas ambient damping turns vertical noise into representative drift at rate γ^-2; horizontal damping suppresses noise but biases the step.The horizontal stochastic error is controlled by the damped Fisher block, with m + γ setting the lower spectral scale for noise amplification.Source: §8.2 · p. 30
  • Gauge-projected Tikhonov updates stay horizontal; along common eigendirections, larger metric penalties shrink update amplitude and shot-noise variance, while noncommuting blocks require Schur-complement treatment.The proposition conditions on pilot information fixing the Fisher estimate, structural penalty, and horizontal chart before gradient shots are drawn.Source: §8.2 · pp. 30–31
  • With estimated projectors, vertical leakage grows with subspace error; an exact projector removes instantaneous vertical shot noise, while a fixed exact projector eliminates cumulative vertical displacement.The perturbation bound can use primitive radii, though that form is more conservative than the estimated-block hypothesis.Source: §8.2 · p. 31

8.3 Identifying physical release directions

  • A physical release direction must have positive Fisher length at the probing state; the released Fisher kernel is then recomputed to determine the horizontal block for damping.Pure-state directions can be tested using F(v, v) = 4 Var(Av); mixed states use the symmetric-logarithmic-derivative Fisher Gram.Source: §8.3 · p. 32

8.4 End-to-end release certificate

  • On one high-probability event, the workflow certifies positive penalized gain, local descent, release-dimension recovery under spectral separation, and a post-selection oracle inequality.The guarantee combines release, validation, and optimization statements under a union bound over gate, release, validation, and projector-estimation events.Source: §8.4 · pp. 32–33
  • Estimated-projector steps satisfy pathwise leakage bounds, but accumulated squared local leakage is not an endpoint-displacement guarantee without a fixed projector or specified transport.Source: §8.4 · p. 33

9 Finite-Sample Releasable Equivariant Learning Algorithm

The algorithm orders release probing, spectral selection, training, and validation across independent data streams. It selects among stored candidates using validation without feeding validation results back into candidate generation.

9 Finite-Sample Releasable Equivariant Learning Algorithm

  • Independent fitting, release, and validation streams keep the data roles separate, and validation does not feed back into candidate generation.The workflow fixes the order in which the gate, spectral test, and validation comparison may see data to preserve the final margin.Source: §9 · pp. 33–34
  • The algorithm estimates loss-corrected curvature and, when needed, first-order benefit before forming the penalized release matrix and selecting its certified positive eigenspace.Source: §9 · p. 33
  • The spectral rule opens directions whose estimated penalized quadratic benefit exceeds the uncertainty radius and treats eigenvalues in [−rA, rA] as ambiguous.It compares benefit with declared costs and a simultaneous uncertainty margin in certified Gram-whitened coordinates.Source: §9 · pp. 33–34
  • For each selected structure, it recomputes the released Fisher kernel and horizontal projector, trains candidate models, then selects among stored predictors on validation data.The returned result includes the selected structure, signal and release intervals, risk bound, and geometric diagnostics.Source: §9 · p. 34

10 Numerical Experiments

The experiments test where hard symmetry hides labels, how global and local release tests behave under finite samples and shots, and how release selection and quotient training respond to noise. Across the tested eight-qubit Ising grids, shared-shadow measurements certify release with substantially fewer shots than the specified scalar estimator, while curvature and validation determine when release is justified.

10.1.1 Exact twirling blind-spot classification

  • Hard spin-flip-invariant readouts stay at accuracy 1/2, while releasing normalized magnetization Mz/n rapidly approaches perfect classification once its signal exceeds shot noise.The simulated sign-test scale is nM; the underlying class states have identical spin-flip twirls.Source: §10.1.1 · p. 34

10.1.2 Quantum-native gate, sector spectroscopy, and the point-null boundary

  • At equal 2000-copy budgets, the twirl–swap estimator’s mean absolute error stays between 0.045 and 0.059 from d = 4 to 128, while the single-copy estimator rises from 0.013 to 0.127.In the S3 test, one shared set of 200,000 outcomes estimates both nontrivial sector masses; at a structured point null, the full Pauli scan has false-release probability 0.003 and power 0.8695, where twirl–swap SNR is 0.247.Source: §10.1.2 · p. 35

10.2 Detection boundaries

  • Dense local release power rises above a constant-order normalized signal, while unknown sparse-direction release and recovery become reliable only above the σ log p/M search scale.The dense simulation illustrates the calibrated test; a mixture argument establishes the lower bound for all tests, and the sparse simulation uses p = 64 candidate directions.Source: §10.2.1 · p. 36, §10.2.2 · p. 37

10.3.1 Task-relative release curvature across the Ising transition

  • For C = −(Mz/L)2 and B = My/2, Ising release curvature is positive on the disordered side, negative on the ordered side, and crosses near criticality.The authors frame curvature as a local task-improvement test, not a stand-alone phase label; large order-parameter fluctuations alone do not determine release.Source: §10.3.1 · pp. 37–38, Figure 5 · p. 38

10.3.2 Executable certification on an Ising state

  • At h = 0.5, the leading positive mode is approximately −0.996 eMz −0.093 eXZ+ZX (λ1 = 0.014), showing that zero diagonal curvature does not imply zero release; certification uses the spectrum, not entries individually.At h = 1.5, all 200 repetitions certified the leading direction at M = 107 for δβ = 0.1 and at M = 106, 107 for δβ = 0.25. The δβ = 0.4 grid gave no certificate, and tested budgets at h = 1.0 and 0.5 certified no release.Source: §10.3.2 · pp. 38–39
  • 104 total shots for shared shadows versus 6.3 × 10^7 for the scalar mode at δβ = 0.25, a factor of 6300 on the tested grids.All 280 mode-comparison repetitions satisfied their confidence radii with zero false releases; the comparison prices these estimators and bias bounds, not all measurement strategies.Source: §10.3.2 · p. 39, Figure 6 · p. 40

10.4 Release selection with finite data and finite shots

  • The spectral selector approaches the true release dimension as shots increase, while selected dimension is conservative when either sample size or shot budget is small.The soft-release simulation selects a smaller optimal penalty as N grows; in the even/odd task, release becomes more frequent as breaking strength or sample size grows, while independent validation suppresses false release under the null.Source: §10.4.1 · p. 41, §10.4.2 · p. 41, §10.4.3 · p. 41

10.5 Quotient QNG and mechanism checks

  • Gauge-projected QNG keeps the pure-gauge coordinate fixed, unlike full-space Tikhonov damping, and its vertical drift grows with projector error while remaining far below that baseline for small errors.Additional mechanism checks show that increasing soft-damping penalties reduces horizontal step amplitude while preserving its sign, with a large penalty approximating retention.Source: §10.5.1 · p. 44, Figure 12 · p. 45, Figure 13 · p. 45

11 Discussion and Conclusion

The discussion frames symmetry relaxation as a sequence of statistical decisions: detect task signal, certify a local enlargement, and validate the trained model. It connects those decisions to measurement access and parameter redundancy, while identifying open extensions to the decision theory.

11 Discussion and Conclusion

  • The norm-dual characterization sets the global signal scale, including the rank factor needed to control all bounded-outcome readouts.The group twirl identifies readout signal removed by a hard constraint, while release gradient and curvature test improvement through available circuit directions.Source: §11 · p. 47
  • On the tested eight-qubit Ising grids, shared Pauli records certify the leading direction with 6300 times fewer shots than the specified scalar estimator.The comparison depends on measurement ensemble, operator structure, and bias bound as well as release-space size.Source: §11 · p. 47
  • Independent validation compares adaptively generated predictors, and a fast excess-risk bound preserves a nested release path's approximation–estimation rate for bounded squared loss.Quotient training removes independently identified state-preserving redundancy before damping physical modes [6].Source: §11 · p. 47
  • Open extensions include minimax copy complexity for unknown composite symmetric nulls, uniform or sequential confidence for continuously adapted probes, and resource improvements for nonlocal curvature.The paper also identifies candidate paths with explicit approximation guarantees as a direction for further work.Source: §11 · p. 47

A Finite-Difference Curvature Estimation

The section develops scalar finite-difference probes as a fallback for estimating release curvature when direct commutator measurement is unsuitable. It accounts for truncation bias and statistical error in the certificate and illustrates the estimate on an eight-site chain.

A Finite-Difference Curvature Estimation

  • Scalar probes are the fallback when the double commutator expands into high-weight observables and the shadow bound becomes vacuous.The generator is fixed, Hermitian, normalized in declared coordinates, and its unitary perturbation must be implementable.Source: A Finite-Difference Curvature Estimation · p. 47
  • A central difference of three independent loss probes cancels odd Taylor terms and estimates −L′′(0) up to a quartic remainder.The finite-shot estimate assumes unbiased probes, independent estimates, and a sub-Gaussian bound as specified in the proposition.Source: A Finite-Difference Curvature Estimation · pp. 47–48
  • The matrix certificate includes statistical error and truncation bias before direction selection, then uses the same operator-norm radius for selection and descent.Empirical eigendirections of the simultaneously estimated matrix may reuse its shots, but adaptively proposed additional observables need a fresh batch or enlarged uniform bound.Source: A Finite-Difference Curvature Estimation · p. 48
  • O(M^-1/4) curvature error is the price of estimating a second derivative from noisy scalar passes, while this sampling task avoids dependence on the double commutator's shadow norm.Nonlocal dynamics still incur circuit-synthesis cost, and fixed-step residual bias is added to the confidence radius.Source: A Finite-Difference Curvature Estimation · p. 48
  • 32 is the exact κB for the L = 8 chain at h = 1; central differences approach it from 29.08 at step 0.2 to 31.95 at step 0.025.The stated intermediate estimates are 31.26 at step 0.1 and 31.81 at step 0.05.Source: A Finite-Difference Curvature Estimation · p. 48

B.1 Twirling blind-spot classification

This appendix experiment tests a released classifier on six-qubit product states whose labels determine the sign of their Z polarization. The hard invariant baseline is fixed at chance accuracy.

B.1 Twirling blind-spot classification

  • The experiment uses six-qubit class-conditional product states with polarization parameter m ranging from 0 to 0.8 in increments of 0.05.For each copy budget, all qubits are measured in the Z basis across independent copies.Source: §B.1 · p. 49
  • The released classifier predicts the sign of empirical magnetization and breaks ties uniformly at random.Each point uses 5000 Monte Carlo repetitions.Source: §B.1 · p. 49
  • The hard invariant baseline has accuracy 1/2.Source: §B.1 · p. 49

B.2 Quantum-native detection

The appendix specifies simulation and executable-certification settings for quantum-native detection, release, and training checks. These passages chiefly document experimental designs and budgets rather than reporting outcome comparisons.

B.2 Quantum-native detection; B.3 Detection boundary

  • The quantum-native detection check mixes rank-at-most-16 Ginibre states with their spin-flip twirls and compares specified twirl–swap, single-copy, spectroscopy, and Pauli-scan budgets.The separate detection-boundary simulation uses Gaussian observations and a norm test over dimensions k = 2, 4, 8, 16, 32.Source: §B.2 · p. 49, §B.3 · p. 49

B.4 Sparse search and post-selection validation

  • The sparse-search simulation tests a one-sided maximum over 64 coordinates, while its post-selection experiment selects training improvements and evaluates them on an independent validation sample.The validation simulation uses unbounded Gaussian labels, so its studentized threshold does not satisfy the bounded-loss hypotheses of corollary 7.7.Source: §B.4 · p. 49

B.5 Ising exact diagonalization

  • The Ising exact-diagonalization setup uses periodic chains of lengths 6 through 14 at fields 0.5, 1.0, and 1.5.Sparse Lanczos diagonalization computes the listed ground-state quantities.Source: §B.5 · p. 49

B.6 Multi-branch release; B.7 Finite-data–finite-shot release

  • The multi-branch and finite-data–finite-shot release checks specify noisy curvature matrices, structural penalties, and thresholding rules across repeated budgets.The finite-shot comparison retains eigenvalues above a sampling margin, alongside naive and soft alternatives.Source: §B.6 · pp. 49–50, §B.7 · p. 50

B.8 Soft release; B.9 Structural-risk toy task

  • The soft-release and structural-risk toy tasks use 14 breaking features, varying sample sizes and penalties, and compare hard and released model dimensions.The soft-release penalties are approximately 1.778, 0.944, and 0.562 at N = 32, 128, and 512, respectively.Source: §B.8 · p. 50, §B.9 · p. 50

B.10 QNG normal forms

  • The QNG normal-form check specifies symmetric, released, and gauge coordinates and evaluates projected training with a seeded release and noisy gradients.The setup starts at (1, 0, 0), seeds b = 0.05 at epoch 30, and runs 120 steps for 400 repetitions.Source: §B.10 · p. 50

B.11 Mechanism checks

  • Mechanism checks set budgets for a barren-plateau filter, a cubic bound, and soft damping of noisy horizontal gradients.The specified settings include a 0.05 estimator threshold and 120 soft-damping epochs.Source: §B.11 · pp. 50–51

B.12 Executable certification on a quantum state

  • The executable certification comparison evaluates shadow and scalar estimators on an eight-site Ising chain using fixed confidence budgets and empirical Bernstein intervals.Known-state quantities audit results after selection but do not enter the confidence radius; the classical cost is excluded from state-shot counts.Source: §B.12 · p. 51

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