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Task-aligned spectral inductive biases in parameterized quantum circuits

Felix Frohnert, Adrián Pérez-Salinas, Sebastian Lange, Marcus Schwaebisch, Evert van Nieuwenburg, Vedran Dunjko

arXiv:2610.23834v1quant-phcs.LG

TL;DR

How PQC architectures can provide task-aligned inductive biases remains poorly understood. This paper encodes a fixed task-defined query alongside a sample-varying signal, connecting query-controlled coefficient modulation to nonuniform Fourier tasks and geometry-derived queries for raw SAR-like classification. Across small-scale spectral and SAR-like experiments, aligned queries improve learning from limited data, while larger-scale practical benefits remain open.

  • Problem

    How practically relevant learning problems match the functional structure of PQCs remains poorly understood, including how to impose task structure governing coefficient combinations.

    Source: §I · p. 1

  • Method

    The paper studies double data reuploading circuits that encode a sample-dependent signal and a fixed, task-defined query whose encoding modulates signal-frequency coefficients.

    Source: §I · pp. 1–2

  • Results

    Small-scale spectral and SAR-like experiments show that aligned queries can improve learning from limited data.

    Source: §V · p. 13

  • Takeaways & Limitations

    Task-prescribed query phases can shape the hypothesis class or function prior without adding sample-dependent information, linking spectral task structure to circuit design.

    Source: §V · p. 12

    The practical benefits of aligned queries at larger scales remain open.

    Source: §V · p. 13

Abstract

from arXiv · show

Parameterized quantum circuits are widely investigated in quantum machine learning, yet how their architecture can provide task-aligned inductive biases remains poorly understood. We study data reuploading quantum circuits with two distinct inputs: a signal that varies between samples and a fixed, task-defined spectral query that specifies how the signal should be probed. We show that encoding the query can change the circuit's hypothesis class over the signal by modulating how its existing signal frequencies are combined, without introducing new sample-dependent information. This mechanism matches tasks built on nonuniform Fourier transforms, whose evaluation frequencies specify the query and whose sampling positions guide its encoding. We examine this connection in a classification task inspired by synthetic aperture radar, where acquisition geometry determines a query for classifying unfocused spectral patches before image formation. Across small-scale spectral and radar-like tasks, aligned queries improve learning from limited data over parameter-matched circuits with mismatched or absent queries. These results provide a route from known spectral task structure to the design and validation of inductive biases in parameterized quantum circuits.

I. INTRODUCTION

The paper studies how fixed task structure can shape the functions parameterized quantum circuits learn, focusing on data reuploading circuits with a signal and a fixed query. It connects query-controlled coefficient modulation to spectral tasks and geometry-aligned classification of raw SAR-like data.

I. INTRODUCTION

  • A principled account of task-aligned inductive biases in practically relevant PQC learning problems remains limited, despite extensive study of PQC universality and trainability.Prior work has emphasized accessible frequencies and their learning, while how to impose task structure governing coefficient combinations remains open.Source: §I · p. 1
  • DDR circuits encode a sample-dependent signal and a fixed, task-defined query in alternating layers; the query can change represented functions without adding sample-specific information.The query specifies Fourier evaluation frequencies in the spectral tasks studied.Source: §I · p. 1, §2 · p. 3
  • The paper derives a query-controlled coefficient mechanism and reports that aligned queries improve limited-data learning on spectral tasks relative to parameter-matched circuits with mismatched or absent queries.The spectral tasks use fixed nonuniform Fourier evaluations; their frequencies supply the query and sampling positions guide its encoding.Source: §I · p. 2
  • For SAR-like classification, acquisition geometry supplies a query for classifying unfocused measurements before image formation; the empirically preferred query tracks the geometry-derived prediction across acquisitions.Supplying the predicted query improves classification when labeled scenes are scarce.Source: §I · p. 2

1. Data reuploading circuits as generalized trigonometric polynomials

  • Data reuploading circuits produce generalized trigonometric polynomials whose encoding determines possible frequencies, while trainable blocks constrain realizable coefficient vectors.Circuits with the same frequency spectrum need not have the same hypothesis class.Source: §1 · pp. 2–3

2. Double data reuploading and query-modulated coefficients

  • At a fixed query, the query modulates coefficients over a fixed set of signal frequencies, with the query encoding shaping the available coefficient matrices.The signal encoding, trainable architecture, and observable are held fixed in defining the hypothesis class over the signal.Source: §2 · pp. 3–4

B. When does the query change the model?

A query can modulate coefficients without adding signal frequencies, but a change in coefficients at fixed parameters does not by itself establish a model-level change. Whether the hypothesis class or function prior changes depends on whether trainable blocks can absorb the query effect.

B. When does the query change the model?

  • A coefficient change at fixed parameters need not change the hypothesis class, because another parameter setting may produce the same function.The analysis distinguishes accessible frequencies, coefficient modulation, and changes to the hypothesis class or function prior.Source: B. When does the query change the model? · p. 4, §2 · p. 4

1. Frequency support and coefficient modulation

  • The query cannot add frequencies to the signal expansion; its effect is instead constrained to coefficient modulation within the existing signal-frequency space.The joint path-frequency set factorizes under disjoint signal- and query-encoding layers with commuting generators within each layer.Source: §1 · p. 4
  • At fixed parameters, the rank of the query-induced coefficient image bounds how many independent signal coefficient vectors varying the query can explore.A one-dimensional image makes the vectors proportional, although their overall scale may still change.Source: §1 · p. 4

2. Query absorption and three possible effects

  • When each query gate can be absorbed into an independently parameterized following block, the hypothesis class is independent of the query value.If block distributions are also right-invariant under the query gates, the function prior is query-independent; absorption is sufficient, not necessary.Source: §2 · p. 4
  • Failure of the absorption test does not prove a class change, because a reparameterization across several layers may still remove the query effect.Product single-qubit blocks can absorb local query gates, while generic entangling gates lie outside that product group.Source: §2 · p. 4
  • Query effects divide into no model-level change, prior reweighting within an invariant class, and a hypothesis-class change; all three retain the same signal-frequency space.The paper targets a class change, which requires failure of the stated absorption condition for at least one layer and query value.Source: §2 · pp. 4–5

C. Numerically distinguishing the three cases

The numerical studies distinguish changes in individual circuit outputs from changes to the function prior or hypothesis class, while connecting the query to fixed nonuniform Fourier tasks. The experiments preserve the accessible frequency sets so that the tested effects concern coefficients rather than frequency support.

C. Numerically distinguishing the three cases

  • The diagnostics separate fixed-parameter function change, class change, and prior change: Δ detects the first, S can certify the second, and nonzero DΣ certifies the third.A sampled positive S is a lower bound on the class witness; DΣ = 0 alone does not establish equality of full priors.Source: C. Numerically distinguishing the… · p. 5
  • The nonlocal query encoding changes the hypothesis class and function prior, while the absorbable local encoding changes neither under a Haar prior and changes only the prior under a non-Haar prior.The comparisons preserve the accessible query-frequency spectrum; the local path remains absorbable, whereas the nonlocal path can entangle the qubits.Source: C. Numerically distinguishing the… · p. 5, Figure 3 · p. 6
  • Changing the query value changes individual outputs even in the local Haar case, but only the nonlocal encoding yields a class change at generic query values.The class witness vanishes where the nonlocal query gates reduce to product unitaries.Source: C. Numerically distinguishing the… · p. 5

1. Spectral classification tasks

  • Spectral classification tasks use target scores depending on prescribed finite nonuniform Fourier evaluations, with labels assigned by a fixed rule from those scores.The score map may be nonlinear, including spectral intensities or interactions between evaluations.Source: §1 · p. 6
  • For these tasks, evaluation frequencies provide the query value, while sampling positions determine the phase modes that guide its encoding.The positions and frequencies are fixed across signals and specified independently of labels.Source: §1 · pp. 6–7

2. Query Fourier structure of spectral scores

The section connects polynomial nonuniform Fourier scores to the query-dependent Fourier structure of double data reuploading circuits. It then separates phase alignment from coefficient error and states a limited-data learning hypothesis for task-aligned queries.

2. Query Fourier structure of spectral scores

  • For polynomial MNUDFT scores, query frequencies are sums and differences of lifted sampling-position modes, while signal-dependent coefficients are polynomials in signal samples and their conjugates.The resulting score and DDR expansion share a separation of query dependence into phases and signal dependence into coefficients.Source: §2 · p. 7
  • When the score map is continuous on a compact set containing the relevant MNUDFT coefficients, these polynomial-form scores uniformly approximate the target score over the corresponding signal and query domain.The passage states this approximation for the specified compact domain.Source: §2 · p. 7

B. Phase alignment and coefficient error

  • The score-error bound separates phase mismatch from coefficient mismatch and contributions from unused circuit modes; controlling both terms suffices for score approximation.At a fixed query, the two expansions can agree without matching term by term.Source: B. Phase alignment and coefficient error · p. 7
  • A constructed query encoding places each target mode within 2^-Lf ||R||2 of a circuit mode, with injective assignments and arbitrarily small phase-error bounds at sufficient depth.A separate construction exactly realizes a phase-only MNUDFT intensity contrast, including its coefficient functions.Source: B. Phase alignment and coefficient error · p. 7

C. Learning hypothesis

  • The learning hypothesis is that a task-aligned query improves limited-data learning against parameter-matched models with mismatched or absent queries.The query and encoding are shared across samples and specified independently of labels, while training must learn signal-dependent coefficients.Source: C. Learning hypothesis · p. 8

D. Teacher–student test of query alignment

In teacher–student experiments, the task-aligned query improves learning over mismatched or absent queries, and the useful-query region extends beyond one exact value. In SAR-like classification, acquisition geometry supplies a fixed query for classifying unfocused spectral patches.

D. Teacher–student test of query alignment

  • Teachers were two-qubit DDR circuits evaluated at f⋆ = (π/4, π/3), and the controls changed only the fixed query while sharing the trainable architecture.The controls used 2f⋆ or zero; the correct-query class contains the teacher by construction, while the zero-query circuit factorizes across signal coordinates.Source: D. Teacher–student test of query… · p. 8
  • The correct-query student reaches higher balanced accuracy with fewer training examples than either control; the mismatched query does not reproduce the gain despite retaining the nonlocal query branch.Figure 4 compares students with the same architecture, parameters, initializations, and optimization budget.Source: D. Teacher–student test of query… · p. 8, Figure 4 · p. 8
  • Binary labels leave the decision boundary unconstrained between samples, and bounded-logit cross-entropy can trade near-boundary errors for larger margins elsewhere.The remaining errors concentrate near the teacher’s decision boundary.Source: D. Teacher–student test of query… · pp. 8–9

E. Sensitivity to query error

  • 0.941 matched-query balanced accuracy exceeded 0.734 at zero query and 0.906 at λ = 2; the nominal maximum was 0.945 at λ = 0.8125.The broad plateau means the teacher query was not a unique optimum in this finite-sample experiment.Source: E. Sensitivity to query error · p. 9, Figure 5 · p. 9
  • δ1% = 0.143 rad marks the mean worst-direction tolerance for staying within one percentage point of matched-query accuracy, with a paired-bootstrap 95% interval of [0.106, 0.233] rad.At larger perturbations, degradation increasingly depends on direction; the scans do not establish an optimum over the full query domain.Source: E. Sensitivity to query error · p. 9, Figure 5 · p. 9

A. Acquisition geometry as a spectral query

  • For fixed acquisition, SAR scores after focusing inherit spectral task structure: acquisition-set nonuniform Fourier frequencies define focusing evaluations, while geometry-dependent phase relations can support classification before image formation.When the focused-image decision map is polynomial in complex amplitudes and their conjugates, the raw-data score has the stated nested Fourier structure.Source: A. Acquisition geometry as a spectral… · p. 10

B. A small-scale SAR-like localization task

  • Scenes vary target range and reflectivity and include four off-range scatterers, 10 dB measurement noise, and shared channel phase; the query uses only the leading mixed term.Higher-order mixed terms remain below 0.35 rad for the studied acquisitions, so the query does not exactly correct every phase in the generated data.Source: B. A small-scale SAR-like localization… · p. 11
  • Before training, geometry-derived correction concentrates the target response more strongly in its azimuth cell than zero or wrong-sign correction under the same Fourier readout.The displayed balanced accuracies use known channel calibration; trained circuits instead receive uncalibrated phases and learn channel compensation and an effective readout.Source: Figure 6 · p. 10, B. A small-scale SAR-like localization… · p. 11

C. Circuit with a geometry-derived query encoding gate

The SAR-like circuit encodes a geometry-derived query through cross-register phase corrections that cannot be removed by adjacent trainable blocks. Across the studied acquisitions, geometry-aligned queries are preferred and improve classification most when labeled scenes are scarce.

C. Circuit with a geometry-derived query encoding gate

  • Each query upload applies a position-diagonal phase correction to measured samples, with the geometry-derived query supplying half the leading bilinear correction per upload.The circuit uses two three-qubit registers and applies the geometry-derived correction through nine commuting cross-register RZZ rotations per upload.Source: C. Circuit with a geometry-derived… · p. 11
  • Because trainable blocks act separately on the azimuth and range registers, they cannot absorb the geometry-derived query's cross-register coupling.The fixed-query circuits have 48 trainable parameters, allowing query values to be compared with the trainable architecture and optimization resources held fixed.Source: C. Circuit with a geometry-derived… · p. 11

D. Results

  • The wrong-sign query performs substantially worse despite matching the geometry-derived query's gate count and operator-Schmidt spectrum.The comparison indicates that the benefit depends on correction direction, not merely the presence of an entangling query gate.Source: D. Results · pp. 11–12
  • Across studied training-set sizes, the geometry-aligned query outperforms wrong-sign and zero-query controls, and modestly beats a learned global query at smaller sizes.At the largest size, the paired 95% bootstrap interval for the accuracy difference from the learned query includes zero; the geometry-derived query is most useful when training data offer the least opportunity to infer its effect.Source: D. Results · p. 12
  • For every studied acquisition, the minimum validation loss on the sampled query grid occurs at the geometry-derived query.Local quadratic interpolation also yields preferred-query estimates close to the geometry-derived prediction, a consistency check because the scenes share the same dispersion relation.Source: D. Results · p. 12
  • The experiment uses small synthetic point-scatterer patches and assumes ideal phase upload and noiseless circuit evaluation.It omits the transmitted pulse, antenna response, platform errors, extended-scene speckle, and the complete focusing pipeline.Source: D. Results · p. 12

V. DISCUSSION AND OUTLOOK

The discussion presents fixed-query coefficient modulation as a way to align quantum-circuit function structure with spectral tasks, and reports benefits in small-scale spectral and SAR-like experiments. Larger-scale practical benefits and extensions to general task classifiers remain open.

V. DISCUSSION AND OUTLOOK

  • A fixed query can shape the hypothesis class or function prior by changing how existing signal Fourier modes are combined, without adding sample-dependent information.Task evaluation frequencies specify the query, while sampling geometry guides its encoding.Source: §V · p. 12
  • Aligned queries improve learning from limited data in small-scale spectral and SAR-like experiments, but practical benefits at larger scales remain open.The discussion calls for larger experimentally acquired phase histories and broader MNUDFT-based classifier studies.Source: §V · p. 13
  • The discussion identifies accessible signal-dependent coefficient functions and their finite-data learning behavior as open issues for general MNUDFT-based classifiers.It also notes that larger-scale practical benefits remain unestablished.Source: §V · p. 13
  • In the SAR-like task, acquisition geometry supplies a query independently of training data and labels, and its leading azimuth–range coupling predicts useful query values across acquisitions.The reported experiment uses small synthetic, phase-only patches, a scalar local correction, ideal input encoding, and noiseless circuit evaluation.Source: §V · p. 13
  • The query framework extends to measurement-based learning tasks that prescribe how samples should be phased and combined, with absorption diagnostics assessing model-level changes before training.The diagnostics test whether the choice changes the hypothesis class or function prior.Source: §V · p. 13

Appendix A: Details on query-controlled inductive bias

The appendix derives Fourier representations for single- and double-input data-reuploading circuits, then establishes how signal frequencies separate and how query variation is bounded by a coefficient-matrix rank.

1. Single data reuploading circuits

  • A single data-reuploading circuit has an expectation value that is a generalized trigonometric polynomial of the input, with accessible frequencies determined by encoding-Hamiltonian eigenvalue spectra.Parameterized unitaries shape the relative weights and phases of spectral components.Source: §1 · p. 17

2. Double data reuploading circuits

  • Double data reuploading alternates signal- and query-dependent encodings with trainable unitaries, yielding a generalized trigonometric polynomial in the query whose coefficients are themselves such polynomials in the signal.The two inputs have separate frequency sets in the joint expansion.Source: §2 · pp. 17–18

a. Proof of spectral separation

  • Because signal- and query-encoding layers are disjoint, every pair of signal and query frequencies occurs in the formal joint frequency set, although restricted architectures may have zero coefficients for some pairs.The factorization concerns formal frequencies, not guaranteed nonzero output terms.Source: a. Proof of spectral separation · p. 18

b. Proof of the modulation bound

  • The query can explore coefficient vectors only within a space whose dimension is bounded by both the rank of C(θ) and the dimension of the query-phase domain.When the query domain contains a nonempty open set, the phase vectors span the full query-frequency space; restricted or discrete domains may yield a smaller space.Source: b. Proof of the modulation bound · p. 18

c. Proof of query absorption

The appendix gives sufficient conditions for query absorption to preserve a circuit’s function class and prior, contrasts alternating encodings with concatenated inputs, and details the query experiments and diagnostics.

c. Proof of query absorption

  • If each query encoding can be absorbed into its adjacent independently parameterized block, the realizable functions of the signal are independent of query value; right-invariant block distributions also preserve the function prior.These are sufficient conditions for invariance under the stated block-independence assumptions.Source: c. Proof of query absorption · p. 18

4. Comparison with concatenated-input encodings

  • Concatenated-input encodings need not generate a Cartesian product of signal and query frequencies, whereas disjoint-register generators provide a sufficient condition for that factorization.With shared-support generators, factorization depends on the joint eigenspectrum and is not guaranteed.Source: §4 · p. 19

5. Details on query experiments

  • The query experiments vary encoding Ef at fixed f or f at fixed encoding, hold signal- and query-frequency sets fixed, attribute observed effects to coefficients alone, and perform no parameter optimization or training.The circuits use two qubits initialized in |++⟩ with L/2 = 2 blocks. Retaining ω0, ω1 ∈ {−2, …, 2} gives the 5 × 5 matrix A; Ng = 8 resolves these modes without aliasing.Source: §5 · pp. 19–20
  • The class witness S is estimated by maximizing over 1,000 parameter samples, so finite sampling can only underestimate it; DΣ > 0 implies a prior change, but DΣ = 0 does not establish full-prior equality.The prior diagnostic uses 5,000 independent parameter samples, jackknife bias correction, and a finite-sample null threshold.Source: §5 · p. 20

a. Varying the query encoding

The appendix’s query experiments show that query encoding and query value can alter circuit outputs and, depending on absorbability and prior, the hypothesis class or function prior. It also establishes a real-coordinate representation for complex spectral tasks and an expansion of polynomial scores over query phases.

a. Varying the query encoding

  • ∆ reaches 0.938, 0.998, and 0.970 at the endpoints of the three query-encoding series; the class witness reaches S = 0.66 on the nonlocal series.The local series have S = 0, while the local narrow-prior series changes the prior despite exact class invariance.Source: a. Varying the query encoding · p. 20

b. Varying the query value

  • The nonlocal query-value series reaches S(t) = 0.695, while both local series remain exactly separable; the narrow-prior local series reaches DΣ = 0.997 despite exact class invariance.The nonlocal class witness has exact zeros at t = 0, π, and 2π, and the Haar local series stays within its 4.0 × 10^-3 null threshold.Source: b. Varying the query value · p. 21

1. Real representation of complex signals

  • A bijective real-coordinate representation preserves all information in a complex signal and preserves the query-dependent Fourier structure of the learning task.Polynomial decision maps in MNUDFT coefficients and their conjugates remain polynomial in the real input components.Source: §1 · p. 21

2. Proof of the MNUDFT score expansion

  • Polynomial decision maps of MNUDFT coefficients yield finite query-frequency expansions, and the continuous case admits uniform approximation over signal and query pairs whose coefficient vectors lie in a compact set.The proof uses polynomial approximation on the compact coefficient-vector set.Source: §2 · p. 21

3. Details of the spectral alignment experiment

The spectral-alignment experiments test whether query encodings can approximate task-defined Fourier modes while remaining active in the chosen circuit. The construction bounds mode-approximation error, and a four-qubit experiment finds a query-induced change in the function prior.

3. Details of the spectral alignment experiment

  • Dyadic query-generator scales halve the worst-case task-mode distance with each layer, whereas uniformly spaced scales converge only polynomially.The dyadic construction keeps the distance below the bound NR 2^-Lf.Source: §3 · p. 22
  • The aligned encoding reaches a normalized second-moment distance of 0.345±0.010, while the absorbable isospectral control remains at the sampling floor.The two encodings have identical generator eigenvalues and signal-frequency support; only the aligned encoding changes the parameter-induced function prior.Source: §3 · p. 22

4. Derivation of the alignment bound

  • The alignment-bound derivation compares target and circuit representations by adding and subtracting a mode expansion, then bounding phase differences and taking a supremum over inputs.The resulting bounds are identified as Eqs. (26)–(28).Source: §4 · p. 22

5. Task-spectrum realizability

  • For any finite target spectrum, Proposition 6 gives query generators whose accessible modes approximate every target mode when 2^-Lf ∥R∥2 ≤ ε.Distinct target modes are assigned injectively when 2ε is below their minimum pairwise separation.Source: §5 · p. 23
  • The target spectrum is determined by sampling positions and the decision map, while the task query supplies the evaluation point used to choose the generators.The construction uses one qubit per query coordinate and covers the coordinate-wise extent of the complete target spectrum, so it establishes existence rather than resource optimality.Source: §5 · p. 23

6. Exact realization of an intensity contrast

The section gives an exact DDR construction for a normalized MNUDFT intensity contrast. It establishes a specific realizable spectral task, not general MNUDFT computation or efficient implementation.

6. Exact realization of an intensity contrast

  • At a parameter value θ⋆, the circuit and target agree for every query pair, with phase and coefficient errors both zero.The construction yields exact containment of the normalized intensity-contrast score in an (m + 1)-qubit DDR hypothesis class.Source: §6 · p. 23
  • The nonlinear decision map is realized exactly, and thresholding its sign gives a binary spectral task.The signal magnitudes are fixed; class information is carried by relative phases and coherent responses at two prescribed nonuniform queries.Source: §6 · p. 25
  • This exact result does not show that general DDR circuits compute MNUDFTs or realize arbitrary spectral decision maps, and its diagonal encodings may require gates growing with D.It is an expressivity result rather than an efficiency claim; for general tasks and restricted ansätze, coefficient error remains a learning problem.Source: §6 · p. 25

7. Details on teacher–student experiments

These experiments and background passages connect nonuniform spectral queries to DDR circuits and describe teacher–student tests of query alignment. The passages explain how acquisition geometry determines SAR queries and how the query gate interacts with the circuit architecture.

7. Details on teacher–student experiments

  • With 32 labels, a student connects regions the teacher separates in unconstrained gaps; with 256 labels, residual errors lie in a thin band along the teacher boundary.The zero-query student's factorized score restricts its decision boundary to straight lines, unlike the teacher's central lobe and boundary-wrapping pockets.Source: §7 · p. 26, Figure A5 · p. 26

1. Background: nonuniform Fourier transforms

  • A multidimensional NUDFT evaluates samples at arbitrary positions and query-frequency vectors, and SAR focusing can be expressed using MNUDFT values followed by fixed operations.Conventional Stolt interpolation approximates the same finite-data spectral values from a regular measurement-frequency grid.Source: §1 · p. 26, §2 · p. 27

2. Nonuniform queries in ω-k focusing

  • SAR image-grid locations map to a non-Cartesian set of measurement-frequency queries fixed by acquisition and the image grid.The MNUDFT of the sampled echoes supplies these spectral values before fixed filtering, weighting, and inverse transformation.Source: §2 · p. 27

3. Local geometric phase and the circuit query

  • The geometry-derived query gate is nonproduct across the two registers and cannot be absorbed into adjacent trainable blocks of the form UA ⊗ UD.Its operator-Schmidt rank is eight when the geometry-derived query values are distinct; the gate decomposes into nine cross-register RZZ rotations.Source: §3 · p. 27

4. SAR-like data generation and training

The SAR-like setup derives a query from acquisition geometry and trains phase-only circuits on sampled patches. The supplied passages describe the circuit, data conditions, and training and evaluation procedure.

4. SAR-like data generation and training

  • The query is calculated from the acquisition geometry, and the generated phase differs from its bilinear approximation by less than 0.35 rad for every acquisition.The target-aligned correction generally leaves clutter with residual mixed phases because clutter scatterers have different geometric phase curvatures.Source: Table A2 · p. 28, §4 · p. 28
  • The six-qubit circuit uses three-qubit azimuth and range registers, phase-only signal input, and 48 trainable parameters.A Fourier readout on the azimuth register yields class probabilities after marginalizing over the range register.Source: §4 · p. 28
  • Training uses full-batch cross-entropy with 700 Adam steps, and paired datasets and parameter initializations are used across compared queries and models.The evaluation compares aligned, wrong-sign, zero, and trainable queries on nested class-balanced training sets; reported accuracies are balanced test accuracies.Source: §4 · pp. 28–29
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