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Fractal dimension predicts quantum kernel collapse in angle-encoded data

Ana Paula Appel

arXiv:2609.00475v1quant-phcs.LG

TL;DR

Angle-encoded quantum kernels on tabular data can collapse before labels are used when circuit width exceeds the data’s intrinsic dimension. The paper proposes D2 with FD-ASE as an a priori width-and-coordinate selection, finding that one-layer ZZ kernels remain alive at the fractal width while PCA-95% widths often collapse, with the ceiling depending on map and bandwidth.

  • Problem

    Angle-encoded quantum kernels on tabular data can collapse before labels are used when circuit width exceeds the data’s intrinsic dimension.

  • Method

    The paper estimates correlation fractal dimension D2 for the qubit budget and uses FD-ASE to select original coordinates for angle encoding.

  • Results

    On nine data sets, one-layer ZZ kernels at the fractal width remain geometrically alive while PCA-95% widths have already collapsed; IBM Quantum matches the live kernel with MAE 0.012–0.021.

  • Takeaways & Limitations

    The usable qubit ceiling is a property of the map–data–bandwidth triple, so ⌈D2⌉ must be reported with the feature map and bandwidth.

  • Takeaways & Limitations

    The evidence is limited to one backend, n = 8 and n = 16 hardware samples, 256 shots, no error mitigation, and selected maps.

Abstract

from arXiv · show

Angle-encoded quantum kernels on tabular data collapse when the feature map is wider than the intrinsic dimension of the data. We propose the correlation fractal dimension D2 as an a priori qubit budget: encode D2 coordinates chosen by FD-ASE instead of the PCA-95% width or all E attributes. On nine data sets and a statevector simulator (n= 32), a one-layer ZZ fidelity kernel at q=D2 stays geometrically alive while the same kernel at the PCA-95% width has already collapsed. The budget is map-dependent: product-state and IQP maps overshoot it; a second ZZ layer undershoots it. Packed dense-angle and re-uploading encodings still live at the fractal q, but not when PCA-95% features are stacked onto those qubits. Shrinking the angle bandwidth moves the ZZ knee later; stretching it kills the kernel earlier. On IBM Quantum (ibm_fez, 256 shots, n=8) the one-layer ZZ kernel at the fractal width matches the exact kernel (MAE 0.021); past that width both hardware and simulator have collapsed. The ceiling is a property of the map-data pair at a stated bandwidth, not of the classical table alone.

1 Introduction

The paper frames quantum-kernel collapse as a mismatch between recorded feature width and the data’s intrinsic dimension, then proposes fractal geometry as an a priori qubit budget and tests it across maps, data sets, and hardware.

  • Motivation: Quantum kernels can collapse before labels are used when angle-encoded states become nearly orthogonal as qubit count grows.The resulting kernel concentrates near the diagonal, erasing similarities needed by downstream kernel methods.
  • Motivation: The recorded attribute count E can exceed the intrinsic dimension of the data, adding redundant rotations that push the fidelity kernel toward a diagonal matrix.A circle embedded in R20 still has one degree of freedom, while padded axes consume additional circuit width.
  • Motivation: PCA-95% can select a width beyond the kernel’s geometric survival point, motivating a classical, a priori estimate of how many coordinates a feature map can bear.The paper contrasts this with selecting qubits by explained variance or post-circuit accuracy.
  • Study design: The study compares fractal-dimension-selected coordinates with all attributes, principal components, and random axes using the same angle maps, fidelity kernel, and downstream algorithms.The experiments cover nine data sets, map variants, robustness checks, and IBM Quantum hardware.
  • Study design: Three kernel diagnostics and an operational alive rule identify the last usable width, called the knee, at n = 32.The knee is compared with ⌈D2⌉ and rival ceilings including TwoNN and PCA-95%.

2 Background

The background distinguishes embedding width from intrinsic dimension, introduces D2 and FD-ASE, and defines map- and bandwidth-sensitive diagnostics for deciding when a fidelity kernel remains geometrically usable.

  • Intrinsic dimension: Intrinsic dimension measures the degrees of freedom of a data support rather than its recorded width, class count, or covariance rank.A Swiss roll has intrinsic dimension two despite requiring three linear components in its embedding.
  • Fractal dimension: D2 estimates intrinsic dimension from the scaling S(r) ∝ r^D2 across shrinking distance scales, with redundant axes contributing little.The estimate is computed classically using multiple scales and unit-cube normalization.
  • Feature selection: FD-ASE ranks attributes by partial fractal dimension and returns original columns, whereas PCA returns linear mixtures of the recorded axes.This distinction determines whether each qubit encodes an interpretable measurement or a blend that may include padding.
  • Angle maps: The default maps independently scale coordinates to [0, cπ], making the bandwidth c a control on angular separation and concentration.The paper uses c = 1 unless otherwise specified and notes that shrinking c retards concentration.
  • Angle maps: Product-state Z maps concentrate more slowly than entangled maps, while a second ZZ layer can be geometrically dead at the fractal width.Packed dense-angle and re-uploading maps test whether multiple coordinates can share fixed wires.
  • Kernel diagnostics: The mean off-diagonal similarity approaches zero when states become pairwise orthogonal, while near ≈ far indicates poor neighbour ranking even with residual off-diagonal mass.The alive rule separates these failure modes using near, far, and mean off-diagonal diagnostics.
  • Kernel diagnostics: For breast cancer, the one-layer ZZ knee is q = 3 = ⌈D2⌉, whereas PCA-95% would encode q = 10.At q = 3, near, far, and mean off-diagonal satisfy all alive clauses; at q = 4 they fail.

3 Related work

Prior work establishes quantum-kernel concentration, dimensionality-reduction approaches, and fractal-dimension feature selection, but does not use an intrinsic-dimension estimator as an a priori qubit budget.

  • Quantum kernels: Quantum feature maps define implicit Hilbert-space kernels, and global feature-map fidelity kernels can concentrate exponentially with qubit count.The cited literature connects quantum models to kernel methods and concentration phenomena.
  • Quantum kernels: Existing effective-dimension and generalization analyses characterize properties of quantum models rather than the support geometry of the classical data.The paper positions its data-geometry budget as distinct from these model-level notions.
  • Dimensionality reduction: Autoencoder-based quantum-classifier reductions still choose qubit counts by trial or PCA variance thresholds.This leaves the qubit-width decision downstream of representation selection.
  • Fractal dimension: Data-mining research developed correlation-fractal-dimension methods for identifying geometrically informative attributes, including the FD-ASE extension.The paper adapts this line of work to angle-encoded quantum kernels.

4 Proposed method

The proposed method separates how many qubits to use from which coordinates to encode: ⌈D2⌉ sets the width, and FD-ASE selects original attributes for the quantum map.

  • How many qubits: The method computes D2 on classical data and sets the qubit ceiling to q* = ⌈D2⌉.The geometry sample can use up to 4000 rows and eight resolution levels.
  • Which coordinates: FD-ASE then selects the coordinates, encoding original columns when at least two are available and otherwise falling back to the first q* principal components.The selection and width are separate decisions in the proposed two-step budget.
  • Comparison: The evaluation keeps the QML model fixed while comparing fractal-selected attributes with every attribute, leading principal components, and random axes.The comparison concerns geometric usability of the quantum kernel.
  • Evaluation scope: The paper defines “better” as retaining a geometrically usable quantum kernel, not as showing that QSVM outperforms SVM.This scope distinguishes the proposed budget from a broad classifier-performance claim.
  • Method rationale: FD-ASE selects a subset of original measurements, while PCA blends signal and padding into each principal component.The paper’s claim is map-specific: fractal width often remains usable where PCA-95% is already collapsed.

5 Experimental protocol

The protocol separates classical geometry estimation from quantum kernel evaluation, using independent samples and a fixed n = 32 diagnostic for comparability. It evaluates competing coordinate views, a one-layer ZZ map, classical controls, and a small IBM Quantum job.

  • Design separation: ⌈D2⌉ sets the number of encoded coordinates, while the QPU sample size n limits how many points can be kernelised.The paper treats qubit reduction as the scientific claim and row reduction as a hardware constraint.
  • Data sets: Nine data sets are evaluated, spanning synthetic low-dimensional embeddings, UCI-style tables, image-like data, pendigits, and a clustered control.The geometry sample contains up to 4000 rows; the kernel sample uses n = 32 rows by default.
  • Sampling: Independent geometry and kernel samples prevent D2 estimation from reusing the small set later sent through the quantum circuit.Wine and diabetes are additionally repeated at n = 128 because they concentrate before the ceiling at n = 32.
  • Kernel protocol: At q = 2, ..., 8, the study compares PCA, random original columns, recorded-column prefixes, and FD-ASE-selected coordinates using a one-layer linear ZZ map.The same coordinate views are also evaluated with a Gaussian RBF control.
  • Hardware validation: IBM Quantum hardware uses ibm_fez with 256 shots and n = 8, producing 28 unordered-pair compute–uncompute circuits per job.The hardware tests one-layer linear ZZ kernels across selected breast, blob, and random-coordinate settings.

6 Results

Across the simulator results, the fractal ceiling tracks the one-layer ZZ kernel’s last alive width better than PCA-95% or TwoNN, while FD-ASE succeeds only when its selected subset remains short enough. The experiments also identify data-set-specific failures and show that a classical RBF control does not exhibit the same collapse.

  • 6.1 Simulator: the ceiling: Breast uses q = 3 at ⌈D2⌉ versus q = 10 for PCA-95%; the fidelity kernel collapses at q = 4, while the fractal-width setting retains usable geometry.At q = 3, QSVM, kNN, and alignment still see geometry; the q = 10 kernel is already diagonal.
  • 6.1 Simulator: the ceiling: TwoNN overshoots moons and intrinsic2 to ≈13 by treating embedding noise as extra dimension, while D2 avoids that inflation.The figure attributes the discrepancy to the 18 noise axes visible at neighbour scale.
  • 6.1 Simulator: the ceiling: The reported failures include OneBig, where the kernel lives past ⌈D2⌉, wine and diabetes, which concentrate before the ceiling at n = 32, and digits, with unreliable box-counting.These are presented as limitations of the proposed ceiling across particular data sets and estimation settings.
  • 6.1 Simulator: the ceiling: ⌈D2⌉ generally sits near the PCA-view knee, whereas PCA-95% often requests 10–40 qubits after the fidelity kernel has died at q ≤5.Figure 1 reports breast on the line at q = 3 and pendigits at 6 versus 5; live knees remain visible at 2–7.
  • 6.2 FD-ASE at its selected width: FD-ASE-selected coordinates are alive on iris, diabetes, and OneBig, but breast, wine, and pendigits show failures when the selected width exceeds the relevant budget or knee.The paper states that FD-ASE realizes the qubit budget when it returns a short subset, not that every selected subset remains alive.
  • 6.2 FD-ASE at its selected width: On breast, fidelity curves fall toward zero as q grows and states become orthogonal, whereas the classical RBF control does not show this collapse.The fidelity-kernel geometry and QML scores peak at q = 3 and degrade beyond the knee.

6.3 Bandwidth

Bandwidth changes the one-layer ZZ kernel’s collapse point, so the fractal ceiling must be reported together with the map’s angle scale. Smaller bandwidth delays collapse, while larger bandwidth accelerates it.

  • c = 0.25 keeps the PCA-view kernel alive through q = 6 on breast, compared with knee q = 3 at c = 1.
  • c = 0.5 leaves the breast knee at q = 3, while c = 2 has no knee in q = 2–8.
  • Shrinking c moves the ZZ knee later and stretching c kills the kernel earlier on breast, moons, and pendigits.
  • The ceiling is therefore map- and bandwidth-dependent, rather than a universal integer determined by the classical table alone.

6.5 Robustness: seeds, floors, and n = 128

Robustness checks show that the breast and moons knee estimates are reproducible across samples, while alive-floor constants and sample size can affect the reported knee. Increasing n helps wine and diabetes but does not reach their fractal ceilings.

  • Seeds: Median last-alive q is 3 for both breast and moons across ten independent n = 32 samples.Breast ranges from 2 to 4, while moons ranges from 2 to 3.
  • Floors: A 50% lower alive-floor setting leaves the breast, moons, pendigits, and intrinsic2 knees unchanged.
  • Floors: Raising the near floor to 0.38 or the ratio floor to 3 can shift the knee by one qubit or eliminate it on breast.
  • n = 128: Increasing n from 32 to 128 moves the wine knee from 2 to 4 and the diabetes knee from 3 to 5, still below ceilings 7 and 6.

6.6 Packed encodings: more than one attribute per qubit

Packed dense-angle and re-uploading maps can encode multiple coordinates per qubit while remaining alive at the fractal width, but stacking PCA-95% width onto those qubits collapses the kernel.

  • At the fractal q, ZZ is alive on breast and moons, whereas ZZ well beyond that width is dead.
  • Dense-angle and re-uploading encodings remain alive at the same q with two PCA coordinates per qubit.
  • Packing 12 PCs onto 3 breast qubits or 16 PCs onto 2 moons qubits fails the alive rule.
  • On moons, PCA stacking makes near/far drop below 1, so distant points appear more similar than neighbours.
  • The fractal width is a qubit budget, not a one-to-one coordinate count, but stacking PCA-95% features cannot recover the extra width.

6.7 Feature-map ablation: does the ceiling follow the map?

The collapse ceiling follows the feature-map construction: one-layer ZZ tracks the fractal ceiling, product-state Z overshoots it, and a second ZZ layer can undershoot it by flattening geometry.

  • One-layer ZZ: One-layer ZZ tracks the fractal ceiling across breast, moons, iris, and pendigits, with last-alive q values 3, 3, 3, and 5.
  • Two-layer ZZ: A second ZZ layer matches some one-layer knees but has no alive width on breast or moons.At breast q = 3, its mean off-diagonal is 0.17, but near and far are 0.25 and 0.22.
  • Product-state Z: Product-state Z remains alive through q = 6 on breast and q = 7 on pendigits, exceeding the corresponding fractal widths.
  • Linear IQP: Linear IQP lies between these behaviours, with last-alive q = 5 on breast, 4 on moons and iris, and 7 on pendigits.
  • Conclusion: Across four data sets, the fractal number is a budget for the intended map rather than a universal integer shared by every circuit.

6.8 Hardware validation

IBM Quantum reproduces the exact one-layer ZZ kernel at the fractal width and beyond it, while FD-ASE preserves near/far ordering in the larger device comparison.

  • At q = 7, the exact and hardware kernels are already concentrated, with hardware agreeing because both are near zero.The n = 8 comparison therefore shows collapse in the exact kernel rather than hardware-induced collapse.
  • The n = 8 jobs use 28 compute–uncompute circuits, so three n = 16 jobs with 120 circuits each are added for near/far scoring.These jobs compare PCA at q = 3, FD-ASE at q = 4, and three random columns at q = 3 on the same breast cancer rows.
  • Packed dense-angle and re-uploading encodings remain alive at the fractal width, but stacking PCA-95%-scale features onto those qubits is grey.The figure identifies stacking extra coordinates onto fewer qubits as not a workaround.
  • At n = 16, FD-ASE is the only view whose hardware kernel ranks neighbours above strangers, with near/far ratio 2.1.PCA and random columns invert the ordering, with ratios 0.8 and 0.5; hardware tracks simulation with MAE 0.023–0.034.

7 Conclusion

The conclusion presents D2 as a map- and bandwidth-specific qubit budget for one-layer ZZ kernels, while showing hardware validation and explicit scope conditions. It also distinguishes qubit reduction from reducing the classical sample used to estimate D2.

  • 7 Conclusion: At q = ⌈D2⌉, the one-layer ZZ fidelity kernel stays geometrically alive, whereas PCA-95% or all E attributes can produce collapsed kernels.Breast is live at q = 3 and dead by q = 4–10; moons is live at q = 2–3 while PCA would encode 19.
  • 7 Conclusion: The budget is map-dependent: a second ZZ layer is dead at ⌈D2⌉, while product-state Z and linear IQP remain alive past it.For breast, the last alive widths are 6 for product-state Z and 5 for IQP.
  • 7 Conclusion: Shrinking the angle scale to c = 0.25 moves the breast knee to 6, while stretching it to c = 2 kills the kernel at every tested q.Bandwidth is therefore part of the map specification rather than a property of the classical table alone.
  • 7 Conclusion: D2 is estimated on up to 4000 classical rows, while the quantum-kernel diagnostic uses n = 32 and IBM tests use n = 8 or n = 16.Thus, fewer qubits does not mean fewer data points; the hardware evidence covers one backend, 256 shots, and no error mitigation.
  • 7 Conclusion: At n = 16, FD-ASE is the only device view that ranks neighbours above strangers, with ratio 2.1; PCA and random columns invert the ordering.This comparison extends the hardware validation beyond the n = 8 reproductions of the exact kernel.
  • 7 Conclusion: Across four data sets, the one-layer ZZ knee at c = 1 tracks the fractal ceiling, with ⌈D2⌉ ranging from 2 to 6.A synthetic k-sweep also shows that D2 recovers known width while the ZZ map dies near three full-range coordinates when k is large.
  • 7 Conclusion: The ceiling is tight for one-layer ZZ at c = 1 but loose for product-state Z or c ≪1, so quoting ⌈D2⌉ without the map and bandwidth would overclaim.FD-ASE is a column selector, not a second ceiling, and is not alive on every table at n = 32.
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