PaperlayerView exact revision on arXiv

Source-linked AI summary

Fourier Analysis of Parametrized Interactive Quantum Classifiers

Fábio Novaes, Fernando M. de Paula Neto, João V. M. Cardoso

arXiv:2609.17991v1quant-phcs.LG

TL;DR

The paper addresses the uncharacterized link between Hamiltonian spectra, open-system reduction, and the Fourier structure of IQCs. It derives their reduced dynamics and uses that analysis to generalize Hamiltonian encodings. Across the considered benchmarks, IQC-Multi has the strongest aggregate classification performance, but model performance depends on task compatibility and global expressibility does not directly predict it.

  • Problem

    How input-dependent environmental eigenvalues and target-Hamiltonian spectra determine the Fourier structure of open-system quantum classifiers had not been explicitly characterized.

    Source: §1 · pp. 3–4

  • Method

    The paper derives the reduced-channel dynamics and Fourier prediction structure of IQCs, then generalizes them with IQC-Multi, whose environmental Hamiltonian can yield multidimensional frequencies.

    Source: §1 · p. 4, §9 · p. 33

  • Results

    Across the considered benchmarks, the IQC-Multi family achieved the strongest aggregate classification performance, but results do not imply a universal model ordering.

    Source: §9 · p. 33

  • Takeaways & Limitations

    Classification depends on compatibility between accessible Fourier structure and the learning task, while global state expressibility and predictive performance capture different properties.

    Source: §9 · p. 33

    Increasing environment dimension does not guarantee better classification, because the appropriate spectral complexity is dataset-dependent.

    Source: §8 · p. 32

Abstract

from arXiv · show

Interactive Quantum Classifiers (IQCs) constitute a family of quantum machine learning models inspired by open quantum systems, in which the interaction between a target qubit and an environment is described by a Hamiltonian. Previous works introduced alternative Hamiltonian parameterizations and showed empirically that they can improve classification performance, but the role of these parameters in the resulting classifier remains poorly understood. In this work, we derive a closed-form expression for the reduced quantum channel generated by a parametrized IQC with a single target qubit. The analytical solution explicitly reveals how the Hamiltonian parameters control the constant, sine, and cosine components of the classifier output, establishing a Fourier interpretation of the induced feature map. This analysis motivates a generalized family of Hamiltonian encodings, including matrix-parameterized environmental Hamiltonians whose Fourier components depend on linear combinations of input features, thereby enabling non-separable Fourier structures. Numerical experiments on synthetic and real-world datasets show that the proposed models can improve classification performance on several nonlinear benchmarks. The generalized matrix encoding achieves the strongest aggregate performance in the evaluated benchmark, while a simpler four-parameter extension often attains comparable performance with substantially fewer trainable parameters. We additionally characterize the generated state ensembles using the standard fidelity-based expressibility measure, finding that global expressibility does not directly predict classification performance. Our results provide an analytical characterization of parametrized Hamiltonians in Interactive Quantum Classifiers and establish Fourier analysis as a useful framework for understanding and designing open-system-inspired quantum learning models.

1 Introduction

The paper addresses the missing explicit account of how input-dependent environmental and target Hamiltonian spectra shape the Fourier structure of open-system classifiers. It develops IQCs as an analytical framework, generalizes their Hamiltonian parameterizations, and evaluates classification and expressibility.

1 Introduction

  • The paper identifies a gap in connecting trainable Hamiltonian spectra, open-system reduction, and the resulting Fourier structure of quantum learning models.Existing work had not explicitly characterized how input-dependent environmental eigenvalues and the target-Hamiltonian spectrum determine that structure.Source: §1 · pp. 3–4
  • IQCs encode input through an input-dependent environmental Hamiltonian, whose spectrum determines accessible prediction frequencies, and obtain predictions by measuring the reduced target state.This provides an analytically tractable setting for deriving reduced dynamics and prediction functions.Source: §1 · p. 4
  • IQC-Multi generalizes Hamiltonian-based IQCs, with independent control of environment dimension and multidimensional frequencies that can combine multiple input features.It recovers previous Hamiltonian-based IQC instances as special cases.Source: §1 · p. 4
  • Across binary and multiclass benchmarks, greater Hamiltonian freedom can improve classification, but no model ordering is universal because performance depends on dataset–Fourier-structure compatibility.The paper also finds that Haar-based global expressibility does not necessarily track classification performance.Source: §1 · p. 4

2 Background

The background establishes how quantum-circuit Hamiltonians shape Fourier models and distinguishes input encoding in evolution from encoding in the initial state. These connections motivate analyzing the spectra of input-dependent interaction Hamiltonians.

2 Background

  • For fixed encoding Hamiltonians, their spectra determine accessible Fourier frequencies, while other circuit parameters determine the corresponding coefficients.The Fourier representation is a finite series over accessible frequencies.Source: §2.1 · pp. 5–6
  • Input dependence can enter through the unitary evolution or through an input-dependent initial state, yielding input-dependent frequencies or coefficients, respectively.Both data-loading strategies are relevant to the models considered in this work.Source: §2.1 · p. 6
  • Prior extensions include multidimensional Fourier series and trainable-frequency models, motivating analysis of how interaction-Hamiltonian spectra determine reduced-dynamics frequencies.In trainable-frequency models, optimization can modify the accessible spectrum as well as Fourier coefficients.Source: §2.1 · p. 6

3 Hamiltonian Formulation of Interactive Quantum Classifiers

The section formulates IQC-Multi through a Hamiltonian interaction between a target qubit and an environment, then derives its reduced-channel form. The environmental spectrum encodes input and selects target rotations; measuring the reduced target state yields predictions.

3 Hamiltonian Formulation of Interactive Quantum Classifiers

  • IQC-Multi provides a Hamiltonian formulation that encompasses previously considered Hamiltonian-based IQCs, while IQC-AIL remains distinct because it encodes input in the initial environment state.The formulation supports analytical characterization of both reduced dynamics and Fourier representation.Source: §3 · p. 7
  • A single target qubit interacts with an ancillary environment; its input-encoded Hamiltonian eigenvalues determine conditional target evolution, after which the environment is discarded and the reduced target is measured.This reduced-channel formulation underlies the subsequent Fourier analysis.Source: §3 · p. 7

3.1 Hamiltonian formulation

  • The target Hamiltonian is parameterized by α0 and α, while W parameterizes the environment Hamiltonian; the “Multi” suffix refers to frequency vectors combining input features.The target direction and environmental parameter matrix provide distinct Hamiltonian controls.Source: §3.1 · p. 7
  • The identity component of the target Hamiltonian contributes only a global phase and does not affect the reduced state or prediction; trace normalization fixes the traceless component’s scale in numerical implementation.The trace-normalized parametrization is used for α0 ≠ 0, while the original IQC with α0 = 0 uses its unnormalized Hamiltonian.Source: §3.1 · p. 8
  • The environment Hamiltonian is diagonal in the computational basis, and its eigenvalues are linear combinations of input features specified by rows of W.Thus W maps the input feature vector to environmental eigenvalues.Source: §3.1 · p. 8
  • When the environment dimension is Ne = 2^ne, each computational-basis state selects a target rotation whose angle is determined by the encoded environmental eigenvalue.The environment therefore acts as a quantum multiplexor.Source: §3.1 · p. 9, Figure 1 · p. 9

3.2 Reduced-channel representation

  • With an initial product state, tracing out the environment yields a target channel with Kraus operators √p_l U_l, and the prediction is a convex combination of expectation-value models.Each component corresponds to an eigenstate of the environment Hamiltonian.Source: §3.2 · pp. 9–10

4 Fourier Representation of IQCs

The section derives an explicit Fourier representation of the IQC prediction from the reduced Bloch-vector dynamics. Environmental Hamiltonian parameters set accessible frequencies, while target and measurement settings determine their coefficients and the constant bias.

4 Fourier Representation of IQCs

  • Each reduced Bloch-vector component is a finite sum of sine and cosine terms with input-linear arguments, so the prediction inherits a finite Fourier structure.The derivation expands the prediction function explicitly in the input features.Source: §4 · pp. 10–11
  • The rows of W determine accessible frequency vectors, while target-Hamiltonian and measurement settings, the initial target state, and environment probabilities determine Fourier coefficients.The environment dimension bounds the number of frequency components that can contribute.Source: §4 · pp. 11–12
  • The measurement operator’s identity component contributes an input-independent constant term, interpretable as a classifier bias.Source: §4 · p. 12
  • The explicit Fourier representation is the paper’s main theoretical result and supports analyzing existing IQC architectures and IQC-Multi parameterizations.The paper then evaluates their empirical performance.Source: §4 · p. 12

5 IQC Models

The section defines several IQC variants through different Hamiltonian and input-encoding choices, then describes the prediction function used in classification experiments. IQC-Multi unifies the Hamiltonian-encoding variants, while IQC-AIL remains distinct.

5 IQC Models

  • The section introduces the IQC variants and specifies the prediction function adopted for classification experiments.The variants distinguish target- and environment-Hamiltonian parameterizations and, for IQC-AIL, the input-encoding strategy.Source: §5 · p. 12, §5.1 · p. 12

5.1 IQC Variants

  • IQC-AIL moves data encoding from the Hamiltonian to the initial environment state while retaining the same target Hamiltonian as the original IQC.Its environment dimension is set to the number of input features for amplitude encoding.Source: §5.1 · p. 13
  • IQC-Multi uses matrix-parameterized environmental Hamiltonians, allowing environment eigenvalues to depend on arbitrary linear combinations of input features.Its Fourier frequency vectors can be multidimensional, unlike the restricted feature-wise frequencies of the other Hamiltonian-encoding models.Source: §5.1 · p. 13
  • IQC-Multi encompasses the Hamiltonian-encoding architectures considered here, with IQC and IQC-α recovered by restricting its environment dimension and parameters.The original IQC fixes α = (1, 1, 1), whereas IQC-α optimizes the target Hamiltonian.Source: §5.1 · p. 13

5.2 Prediction Function and Optimization

  • IQC, IQC-α, and IQC-Multi have Fourier-like prediction functions, whereas IQC-AIL has a normalized homogeneous quadratic decision boundary.For IQC-AIL, the boundary is constrained by diagonal ˜W and |˜w_l| ≤ 2/3.Source: §5.2 · p. 15

6 Experimental Methodology

The experiments compare six IQC variants across synthetic and real-world classification benchmarks using repeated train/test partitions and two optimization procedures. They also evaluate architecture-level state-ensemble expressibility with a fidelity-based measure.

6.1 IQC Settings

  • All variants share an open-quantum-system-inspired binary-classification framework and differ in Hamiltonian or input-encoding parameterization.Training used labels {−1, +1} and mean squared error as the objective.Source: §6.1 · p. 15

6.3 Datasets

  • Six IQC variants were evaluated on seven synthetic binary benchmarks and four real-world benchmarks, including two binary and two multiclass datasets.Each dataset-model combination used ten independently seeded train/test partitions.Source: §6.2 · p. 16, §6.3 · p. 16

6.2 Experimental Protocol

  • Features were min–max scaled to [0, 1] using only each training partition, then applied unchanged to its test partition.Binary datasets used 80%/20% train/test splits; multiclass splits were class-stratified, and multiclass classification used one-vs-rest with one independently trained IQC per class.Source: §6.2 · p. 16, §6.3 · p. 18

6.4 Optimization Settings

  • Training compared Adam-based gradient optimization with global-best particle-swarm optimization using the same MSE objective.Adam ran for 1000 steps; PSO ran for up to 500 iterations and stopped early when training loss fell below 10^-3.Source: §6.4 · p. 18

6.5 Evaluation Metrics

  • Classification results report accuracy, macro-averaged F1-score, precision, and recall as mean ± standard deviation over ten partitions.The reported classification metrics are macro-averaged for F1-score, precision, and recall.Source: §6.5 · p. 19

Expressibility.

  • Expressibility is evaluated by comparing sampled global-state fidelity distributions with the Haar-random reference using KL divergence.For each model, 10,000 independent state pairs were generated, and the empirical distribution used 75 uniformly spaced bins over [0, 1].Source: Expressibility · p. 19
  • The expressibility distribution characterizes an architecture’s state ensemble under the specified input and parameter sampling, rather than states reached by particular trained parameters.Relative expressibility normalizes against the corresponding idle ensemble to compare parameterizations with different Hilbert-space dimensions and input-encoding mechanisms.Source: Expressibility · pp. 20–21

7 Results

Across eleven datasets, IQC-Multi-8 has the strongest aggregate performance, but model rankings vary by dataset. Statistical tests, decision-region plots, and expressibility analyses further show that performance and global state-ensemble expressibility capture distinct properties.

7.1 Overall performance

  • Dataset rankings vary: IQC-Multi-4 reaches 0.996 accuracy on XOR, while IQC-AIL leads Wine with accuracy 0.939 and F1-score 0.941.On Moons, IQC-Multi-8 reaches 0.929 accuracy, compared with 0.881 for IQC-Multi-4 and 0.856 for IQC-AIL.Source: §7.1 · pp. 22–23
  • Accuracy substantially exceeds F1-score on Caesarian and Pima Diabetes, so accuracy alone may not characterize performance under class imbalance.Figures 2 and 3 visualize accuracy and F1-score across the eleven datasets and distinguish binary from multiclass datasets.Source: Figure 2 · p. 22, Figure 3 · p. 22, §7.1 · p. 23
  • IQC-Multi-8 achieves the highest eleven-dataset mean for accuracy (0.863), F1-score (0.833), precision (0.860), and recall (0.852).The aggregate ranking does not make it optimal on every dataset.Source: Table 3 · p. 23, §7.1 · p. 23

7.2 Statistical Analysis

  • Significant model differences occur for 7 datasets in accuracy, 7 in F1-score, 6 in precision, and 7 in recall; none occur on the three Blobs datasets or Caesarian Section.Most pairwise comparisons for IQC-Multi-8 are not statistically significant, despite its generally high mean performance.Source: §7.2 · pp. 23–24

7.3 Decision regions

  • Multidimensional variants produce richer decision regions that more closely follow Circles and Moons geometry, and alternating regions that separate XOR's four clusters.These regions show the superposed prediction function, not individual Fourier components; their spatial structure is consistent with the multidimensional frequencies enabled by the environment Hamiltonian.Source: §7.3 · pp. 24–25

7.4 Expressibility

  • Fidelity-distribution KL values across different Hilbert-space dimensions are not a dimension-independent architecture ranking, and global expressibility does not directly measure classification performance.The expressibility measure characterizes the global state ensemble, whereas classification depends on the subset reached through training that is useful for a particular dataset.Source: §7.4 · pp. 26, 30
  • When Ne = Nf, relative expressibility stays approximately constant as feature count increases; for IQC-Multi with fixed Ne, it increases with Nf.The IQC-Multi increase occurs without enlarging the Hilbert space and is associated with incorporating additional input dimensions into the environmental encoding.Source: §7.4 · pp. 27, 29
  • At Nf = 8 and common global dimension d = 16, IQC-Multi-8 has the smallest KL divergence, followed by IQC-AIL, IQC-α, and IQC.Figure 9 compares the four models with the same Haar reference distribution.Source: §7.4 · p. 29, Figure 9 · p. 29

8 Discussion

Classification performance depends on how each model’s accessible Fourier structure matches the dataset, rather than following a simple hierarchy of increasing expressivity. The discussion connects that result to the distinct spectral controls of the Hamiltonian parameterizations and cautions that global state expressibility is not a direct predictor of classification performance.

8 Discussion

  • IQC-Multi has the strongest aggregate performance among the considered parameterizations, with IQC-Multi-8 highest, but results do not support universal superiority; benefits depend on spectral compatibility with the task.Several pairwise differences are not statistically significant.Source: §8 · pp. 30–31
  • IQC-α primarily changes Fourier coefficients while preserving feature-wise frequencies, whereas IQC-Multi permits linear-combination environmental eigenvalues and non-separable multidimensional frequencies.The distinction is coefficient control versus spectral-structure control.Source: §8 · p. 31
  • The target Hamiltonian direction determines Fourier coefficients and its effective scale rescales frequencies, with coefficient sharing across environmental components constrained by the normalized target direction.Multiple target qubits could provide additional Fourier-coefficient control.Source: §8 · pp. 31, 33
  • At Nf = 8 and d = 16, IQC-Multi-8 has the smallest KL divergence among models sharing global dimension; when Ne = Nf, relative expressibility remains approximately constant as features increase.The latter regime also increases Hilbert-space dimension, so rising KL divergence does not imply a substantial change in normalized distance from the Haar ensemble.Source: §8 · pp. 31–32
  • Global state-ensemble expressibility does not directly predict classification performance, and the current measure may capture global degrees of freedom absent from the reduced target state used for prediction.The discussion proposes reduced-state or observable-based expressibility measures as a closer reflection of the classifier’s effective function class.Source: §8 · pp. 32–33
  • Different accessible frequencies and their coefficients produce qualitatively different decision boundaries; multidimensional variants show richer spatial structures consistent with their richer Fourier structure.IQC-Multi’s Fourier superposition explains wave-like boundaries, while IQC-AIL’s quadratic function explains cone-like decision lines.Source: §8 · p. 32
  • IQC-Multi independently controls feature and environment dimensions, but greater spectral complexity does not guarantee better classification because suitable complexity is dataset-dependent.The environment dimension primarily controls the number and structure of accessible frequencies, while the target Hamiltonian controls their coefficients.Source: §8 · pp. 32–33
  • IQC-AIL belongs outside the Hamiltonian hierarchy because it encodes input in the initial environment state; the interplay of state-based and Hamiltonian-based encodings remains for future study..Source: §8 · p. 32

9 Conclusion

The conclusion analytically connects IQCs to Fourier representations, separating the environmental Hamiltonian’s role in input-dependent spectral structure from the target Hamiltonian’s role in observable contributions. Across considered benchmarks, IQC-Multi has the strongest aggregate classification performance, while expressibility does not directly predict performance.

9 Conclusion

  • IQCs admit an explicit Fourier expansion, with the environment setting input-dependent spectral structure and the target Hamiltonian controlling how those components contribute to the output.The paper derives target dynamics analytically by treating the IQC as a Hamiltonian-induced reduced quantum channel.Source: §9 · p. 33
  • IQC-Multi achieved the strongest aggregate classification performance across the considered benchmarks, but the results establish no universal model ordering.Performance depends on compatibility between accessible Fourier structure and the learning task.Source: §9 · p. 33
  • Global state-ensemble expressibility and predictive performance capture different properties; richer parametrizations can approach the Haar reference without necessarily improving classification.Task-relevant Fourier expressivity depends on accessible frequencies and control of their associated coefficients.Source: §9 · p. 33
  • Future work includes independently parameterizing frequency scale and target direction, increasing target-qubit count, and extending the formulation to scalable, hardware-efficient implementations.The paper also proposes Fourier-controllability and task-relevant expressibility measures based on reduced target states or observable outputs.Source: §9 · p. 34

A Classification Results

The section presents complete training results per metric and model, with separate tables for binary and real-world classification datasets. The supplied passages identify the tables’ scope but do not report their cell values or comparative outcomes.

A Classification Results

  • Tables 5 and 6 summarize IQC-model results on binary and real-world classification datasets, respectively.The section says complete training results are presented per metric and model; the supplied passages do not give the values.Source: A Classification Results · p. 34, Table 5 · p. 35, Table 6 · p. 36
  • Table 5 reports results for IQC models on binary classification datasets.The caption specifies mean ± standard deviation; no cell values are included in the supplied passage.Source: Table 5 · p. 35
  • Table 6 reports results for IQC models on real-world classification datasets.The caption specifies mean ± standard deviation; no cell values are included in the supplied passage.Source: Table 6 · p. 36

B Statistical Tests

The section reports pairwise Wilcoxon signed-rank tests with Holm correction for F1-score across seven datasets. Each cell gives an adjusted p-value, with arrows indicating significant direction at α = 0.05.

B Statistical Tests

  • Table 7 reports Holm-corrected pairwise Wilcoxon signed-rank tests for F1-score on Circles.Cells give adjusted p-values; arrows indicate significantly higher or lower F1-score, and unmarked values indicate no significant difference at α = 0.05.Source: Table 7 · p. 37, B Statistical Tests · p. 37
  • Table 8 reports Holm-corrected pairwise Wilcoxon signed-rank tests for F1-score on Stripes.Cells give adjusted p-values; arrows indicate significantly higher or lower F1-score, and unmarked values indicate no significant difference at α = 0.05.Source: Table 8 · p. 37, B Statistical Tests · p. 37
  • Table 9 reports Holm-corrected pairwise Wilcoxon signed-rank tests for F1-score on Iris.Cells give adjusted p-values; arrows indicate significantly higher or lower F1-score, and unmarked values indicate no significant difference at α = 0.05.Source: B Statistical Tests · p. 37, Table 9 · p. 38
  • Table 10 reports Holm-corrected pairwise Wilcoxon signed-rank tests for F1-score on Moons.Cells give adjusted p-values; arrows indicate significantly higher or lower F1-score, and unmarked values indicate no significant difference at α = 0.05.Source: B Statistical Tests · p. 37, Table 10 · p. 38
  • Table 11 reports Holm-corrected pairwise Wilcoxon signed-rank tests for F1-score on XOR.Cells give adjusted p-values; arrows indicate significantly higher or lower F1-score, and unmarked values indicate no significant difference at α = 0.05.Source: B Statistical Tests · p. 37, Table 11 · p. 39
  • Table 12 reports Holm-corrected pairwise Wilcoxon signed-rank tests for F1-score on Pima Indians Diabetes.Cells give adjusted p-values; arrows indicate significantly higher or lower F1-score, and unmarked values indicate no significant difference at α = 0.05.Source: B Statistical Tests · p. 37, Table 12 · p. 39
  • Table 13 reports Holm-corrected pairwise Wilcoxon signed-rank tests for F1-score on Wine.Cells give adjusted p-values; arrows indicate significantly higher or lower F1-score, and unmarked values indicate no significant difference at α = 0.05.Source: B Statistical Tests · p. 37, Table 13 · p. 40

Related papers

Loading 2609.17991v1…