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Diversity-Aware Ensemble of Quantum-Classical Models via Combinatorial Fusion

Jingyan Xu, Yuanhong Wu, Samuel Yen-Chi Chen, Sean Patten, Christina Schweikert, D. Frank Hsu

arXiv:2610.13808v1quant-phcs.LG

TL;DR

VQC architecture search typically selects a single circuit and leaves other candidates unused, despite the potential value of model diversity. This paper applies CFA to select and fuse diverse quantum and classical models; across four datasets, the resulting combinations outperform the strongest individual models and relevant ensemble and quantum approaches.

  • Problem

    Quantum architecture search selects a single design and discards other candidates, while existing quantum ensembles rely on simple averaging or ad hoc weights rather than diversity-guided selection.

    Source: §I · p. 1

  • Method

    The paper applies CFA to a pool of quantum circuits and classical models, selecting a diverse subset using cognitive diversity and combining predictions through score- and rank-based fusion.

    Source: §I · p. 2, §VIII · p. 10

  • Results

    Across four datasets, CFA-based fusion consistently outperforms the best individual models and recent state-of-the-art ensemble and quantum approaches; on Ionosphere, accuracy rises from 0.9434 to 0.9811.

    Source: §VII · p. 9, §VIII · p. 10

  • Takeaways & Limitations

    The best combinations across the four datasets include both quantum and classical models, with their diversity harnessed through CFA for performance gains beyond either model type alone.

    Source: §VIII · p. 10

Abstract

from arXiv · show

Variational quantum circuits (VQCs) are highly sensitive to circuit architecture, yet most quantum machine learning pipelines select a single best design and discard the remaining candidates. This paper addresses whether the structural diversity across quantum circuit search spaces can be used to produce stronger predictions through model fusion. We apply Combinatorial Fusion Analysis (CFA), a model fusion framework that selects and combines multiple scoring systems using rank-score function (RSF) and cognitive diversity (CD), to a pool of 29 architecturally diverse classical models and quantum circuits. Using a diversity-aware model selection method, we identify subsets of maximally diverse models and combine them with both score and rank combination. Experiments on four binary classification benchmarks covering different domains demonstrate that CFA-based hybrid classical-quantum fusion consistently outperform the best individual model on every dataset considered with improvements up to 6.03% in accuracy. When comparing with relevant literatures, the CFA framework further achieves superior performance across all four datasets.

I. INTRODUCTION

VQC performance depends on circuit architecture, while quantum architecture search typically selects one design and discards other candidates. This paper applies CFA to select diverse quantum and classical models for fusion and evaluates the approach across four binary-classification datasets.

I. INTRODUCTION

  • Quantum architecture search can discard useful information by selecting one circuit, while quantum ensembles often lack a formal diversity criterion for choosing and combining models.CFA uses cognitive diversity to guide selection and combines the predictions of diverse quantum and classical models.Source: §I · pp. 1–2
  • The paper applies Combinatorial Fusion Analysis to quantum machine learning, selecting diverse scoring systems and combining their predictions.The authors hypothesize that this fusion can outperform the best individual model in the pool.Source: §I · pp. 1–2
  • The evaluation covers four binary-classification datasets from medical diagnosis, signal processing, and cybersecurity, and tests score, rank, and weighted fusion strategies.Model selection considers both cognitive diversity and model performance.Source: §I · p. 2

A. Variational Quantum Circuits for Supervised Learning

  • A supervised-learning VQC encodes classical features, applies trainable variational layers, and measures observables within a pipeline that can be trained end to end.Data re-uploading at each layer extends the representable Fourier spectrum from degree 1 to degree L for L layers.Source: A. Variational Quantum Circuits for… · p. 2
  • Scaling VQCs is challenged by gradients that can vanish exponentially with system size in deep or randomly initialized circuits, alongside NISQ hardware noise and depth constraints.The encoding strategy also strongly affects which functions a VQC can represent.Source: A. Variational Quantum Circuits for… · p. 2

B. Quantum Architecture Search

The related work motivates retaining and combining circuit candidates rather than selecting a single architecture, and frames CFA as a diversity-aware alternative to conventional ensembles. The study evaluates the approach on four binary-classification datasets spanning three domains.

B. Quantum Architecture Search

  • Differentiable quantum architecture search learns architecture weights and circuit parameters jointly, but ultimately returns one architecture and discards the other candidates.Earlier evolutionary and reinforcement-learning search approaches are described as scaling poorly or being sample-inefficient and sensitive to reward design.Source: B. Quantum Architecture Search · p. 2

C. Ensemble Methods and Diversity

  • Ensemble benefits depend on diversity, yet standard methods do not optimize diversity during model selection and quantum ensembles typically use averaging without a formal criterion.The cited ambiguity decomposition expresses ensemble error as average individual error minus average pairwise diversity.Source: C. Ensemble Methods and Diversity · pp. 2–3

D. Combinatorial Fusion Analysis

  • CFA uses rank-score functions and cognitive diversity, defined as their ℓ2 distance, and identifies good individual performance and pairwise diversity as joint criteria for effective fusion.A stated theorem says rank combination outperforms score combination under sufficiently high cognitive diversity.Source: D. Combinatorial Fusion Analysis · p. 3

E. CFA versus Standard Ensemble Methods

  • Unlike standard ensembles, CFA operates in score and rank spaces and filters models by individual quality and pairwise dissimilarity before combining them.The passage states that rank combination outperforms score combination under high cognitive diversity.Source: E. CFA versus Standard Ensemble Methods · p. 3

III. DATASETS

  • The evaluation uses four binary-classification benchmarks—heart disease, Ionosphere, Pima Indians Diabetes, and Spambase—with 270–4601 samples and 8–57 features.The datasets span medical diagnosis, signal processing, and cybersecurity; preprocessing includes a stratified 70-15-15 train-validation-test split.Source: §III · p. 3

IV. QUANTUM CIRCUITS ARCHITECTURE

The model pool includes circuits from a structured quantum architecture search space and curated architectures. The search configurations vary initialization, input-encoding rotation, and entanglement topology, with more expressive parameterized gates than the original framework.

IV. QUANTUM CIRCUITS ARCHITECTURE

  • The circuit pool combines 12 configurations from a custom DiffQAS-QRL variant with eight curated architecturally diverse circuits.Each circuit is trained and evaluated within a unified classification pipeline.Source: §IV · p. 3

A. Variational Quantum Circuits: a general structure

  • A VQC applies a parameterized unitary to classical inputs on n qubits, with trainable rotation angles optimized by gradient descent.The circuit encodes features, applies variational layers with rotations and entangling gates, and reads out Pauli-Z expectation values.Source: A. Variational Quantum Circuits: a… · p. 4

B. Data re-uploading

  • All 12 search configurations use data re-uploading, adding the input projection to rotation angles at every layer; layer count is set to ⌈nfeatures/nqubits⌉.Each layer has its own block of trainable parameters.Source: B. Data re-uploading · p. 4

C. Circuit Architecture Search Space

  • The search framework tests each circuit as a fixed classifier, replacing single-axis rotations and fixed CNOT gates with Rot and parameterized CRZ gates.The expanded gates increase variational parameters per layer from one angle per qubit to three; fixing the rotation gate reduces the design space from 36 configurations to 12.Source: C. Circuit Architecture Search Space · p. 4

1) The Three-Component Search Space:

  • Each configuration combines a Hadamard or no-Hadamard initialization with an RX, RY, or RZ input-encoding rotation.The initialization places qubits in equal superposition or leaves them in the standard computational state; the rotation axis determines how input information is oriented before later operations.Source: 1) The Three-Component Search Space · p. 4
  • The entangling topology is either a brick-wall pattern of parameterized CRZ gates on even- then odd-indexed pairs or a ring connecting each qubit to the next.The ring closes the connection from the last qubit back to the first.Source: 1) The Three-Component Search Space · p. 4

2) Circuit Structure:

The 12 circuits are defined by combinations of three design choices, and each is trained independently as a scoring system for CFA.

2) Circuit Structure:

  • The 12 configurations combine initialization, encoding rotation, and entanglement topology, and each independently trained circuit contributes prediction scores to the CFA pool.Configuration names follow the pattern {H|noH} {RX|RY|RZ} {std|cyc}.Source: 2) Circuit Structure · p. 4

D. Curated Quantum Circuits

Eight curated quantum circuits add architectures distinct from the 12 search-space circuits, spanning different gate primitives and design philosophies. They share a common hybrid wrapper for binary classification.

D. Curated Quantum Circuits

  • Eight architecturally distinct curated circuits supplement the 12 search-space circuits, with a common wrapper for binary classification.The circuits use an input projection and tanh activation, a VQC, a second tanh activation, and an output projection to two logits.Source: D. Curated Quantum Circuits · pp. 4–5

1) StrongEnt (StronglyEntanglingLayers):

  • StrongEnt applies per-qubit Euler rotations followed by a variable-range CNOT ladder that entangles all qubit pairs over successive layers.The control–target range increments each layer and cycles modulo n.Source: 1) StrongEnt (StronglyEntanglingLayers) · p. 5

2) BasicEnt (BasicEntanglerLayers):

  • BasicEnt uses one RX rotation per qubit followed by ring CNOTs, trading expressibility for shallow depth and reduced susceptibility to barren plateaus.Each qubit i connects to (i+1) mod n.Source: 2) BasicEnt (BasicEntanglerLayers) · p. 5

3) TwoDesign (Simplified Two Design):

  • TwoDesign alternates CZ brickwork gates with two trainable RY rotations per qubit, forming an approximate unitary 2-design.It is the pool’s only circuit using CZ rather than CNOT for entanglement.Source: 3) TwoDesign (Simplified Two Design) · p. 5

4) Random (RandomLayers):

  • Random fixes a seeded random gate-wiring pattern at initialization and trains only rotation angles, with independently randomized connectivity in each layer.Its stochastic architecture complements the deterministic circuits.Source: 4) Random (RandomLayers) · p. 5

5) QAOA (QAOAEmbedding):

  • QAOA alternates RX mixer rotations with pairwise MultiRZ Ising-type ZZ cost interactions, with data re-uploading built into the template.No external re-uploading loop is needed.Source: 5) QAOA (QAOAEmbedding) · p. 5

6) TTN (Tree Tensor Network):

The curated pool includes TTN, MPS, and IQP circuits, while a shared hybrid pipeline standardizes classification across VQCs. The broader benchmark includes nine classical baselines alongside 20 quantum classifiers.

6) TTN (Tree Tensor Network):

  • TTN recursively pairs qubits using RY–CNOT–RY blocks, forming a hierarchical contraction, and encodes input data only at the first layer.The hierarchy mirrors coarsegraining in condensed-matter tensor networks.Source: 6) TTN (Tree Tensor Network) · p. 5

7) MPS (Matrix Product State):

  • MPS sweeps the TTN two-qubit block left to right, reducing two-qubit gate depth but limiting long-range entanglement; input is encoded once.Distant-qubit correlations must propagate through intermediate sites.Source: 7) MPS (Matrix Product State) · p. 5

8) IQP (IQPEmbedding):

  • IQP uses a fixed diagonal encoding of linear and pairwise-product feature terms, followed by a trainable StronglyEntangling block, making it the pool’s closest analogue to a quantum kernel method.The IQP encoding is the pool’s only non-trainable encoding.Source: 8) IQP (IQPEmbedding) · p. 5

E. Unified Classification Architecture

  • Every VQC uses the same hybrid pipeline: an input projection maps features to qubits, the circuit returns measurements, and an output projection produces two class logits.The projections are wrapped in tanh, and models are trained with Adam and cross-entropy loss.Source: E. Unified Classification Architecture · p. 5, Figure 3 · p. 6

V. CLASSICAL BASELINE MODELS

  • The framework adds nine classical models—seven standard machine-learning estimators and two deep-learning models—to 20 quantum classifiers.Together, these model families form the quantum fusion benchmark pool.Source: §V · p. 6

A. Classical machine learning models

The classical baselines include seven standard machine-learning estimators and two deep-learning models. Their differing mechanisms provide performance baselines and increase diversity in the fusion pool.

A. Classical machine learning models

  • Seven scikit-learn baselines span boosted trees, random forests, RBF SVM, logistic regression, KNN, and AdaBoost, with each model’s probability estimate or decision-function sigmoid serving as its score.The set also includes XGBoost, GBM, and other tree-based methods, alongside kernel and linear classifiers.Source: A. Classical machine learning models · p. 6
  • Classical models provide strong baselines and add diversity because their mechanisms produce structurally different rank-score functions.The mechanisms range from kernels and boosted trees to neural networks and quantum circuits.Source: A. Classical machine learning models · p. 6

B. Classical deep learning models

  • The deep-learning baselines are an MLP and Tabular CNN, trained with Adam and cross-entropy under the same epoch count and batch-size schedule as the quantum circuits.The MLP uses three hidden layers; the CNN treats features as a single-channel sequence and uses convolutional blocks with adaptive pooling.Source: B. Classical deep learning models · p. 6
  • The Tabular CNN’s convolutional channels and fully connected hidden dimension are matched to the number of circuit qubits; with eight qubits, each dimension is eight.This parameter matching puts the deep-learning model at a comparable parameter scale to the VQCs.Source: B. Classical deep learning models · p. 6

VI. CFA FRAMEWORK

CFA combines scoring systems through diversity-aware selection and combination. Its framework represents each system with a rank-score function and measures dissimilarity between systems using cognitive diversity.

VI. CFA FRAMEWORK

  • CFA provides a workflow for diversity-aware model selection and combination of multiple scoring systems.The framework includes score and rank combination methods.Source: §VI · p. 6

A. Score function, rank function and rank-score function

  • A scoring system assigns numerical scores to data items; sorting by those scores yields ranks, and the rank-score function maps rank positions to sorted score values.The score function carries cardinal information, the rank function ordinal information, and the rank-score function bridges the two.Source: A. Score function, rank function and… · pp. 6–7

B. Cognitive diversity and diversity strength

  • Cognitive diversity measures dissimilarity between scoring systems by comparing their rank-score profiles, while diversity strength averages a system’s cognitive diversity against the others.This differs from correlation measures and prediction-disagreement measures.Source: B. Cognitive diversity and diversity… · p. 7

C. Diversity-aware model selection

Diversity-aware selection filters weak models, identifies highly dissimilar prediction pairs, and adjusts the resulting candidate set to a target size using AUROC. The selection is category-agnostic, allowing classical and quantum models to compete on diversity and performance.

C. Diversity-aware model selection

D. Combination framework

CFA combines selected models in score or rank space using alternative weighting schemes. The framework evaluates these combinations against individual models and reported literature results.

D. Combination framework

  • CFA defines six variants by crossing score- or rank-space fusion with average, diversity-strength, or performance weighting.The framework therefore offers parallel score- and rank-combination strategies with three weighting schemes apiece.Source: D. Combination framework · p. 8
  • The workflow trains VQCs, curated quantum circuits, classical models, and deep-learning models, then selects five using validation-set cognitive diversity and individual performance.Source: D. Combination framework · p. 8
  • Table II reports the strongest individual model and best CFA combination for each of four benchmarks, including combination type and relative improvement.Source: Table II · p. 9
  • Combined models are evaluated using Accuracy and AUROC and compared with the five selected base models and recent literature results.Source: D. Combination framework · p. 9

VII. RESULTS

CFA-combined models outperform the strongest individual model across all four datasets, with dataset-specific model selections and rank–score patterns reflecting diversity among the candidates.

VII. RESULTS

  • Figure 6’s rank–score curves differ across models, especially between VQCs and classical models, reflecting distinct scoring behavior that CFA treats as cognitive diversity.Source: §VII · p. 9
  • The selected subsets differ by dataset: each includes at least one quantum circuit, none selects the same model across all datasets, and the circuit family varies.The mix shifts from a tensor-network circuit for heart disease to hardware-efficient circuits for Pima diabetes and an Ising-inspired QAOA circuit for Spambase.Source: §VII · p. 9, Table III · p. 9
  • On heart disease, the best combined model reaches AUC 0.9444 versus the strongest individual model, TTN, at AUC 0.9106—a 3.38% relative improvement under ASC or WSCP.The optimal combination includes classical and quantum models.Source: §VII · p. 9
  • Ionosphere accuracy increases from 0.9434 to 0.9811, Pima accuracy from 0.7845 to 0.8448, and Spambase accuracy from 0.9349 to 0.9407, a 0.58% gain.Source: §VII · pp. 9–10
  • On heart disease, the proposed method’s AUC of 0.9444 exceeds the reported results from Qi et al. (0.8799), Mukherjee et al. (0.8985), and Heidari et al. [68] (0.9164).Source: §VII · p. 10
  • Because the baselines include ensemble and quantum-based methods, the authors argue that gains reflect CFA’s systematic search and selection, not merely using more models.Source: §VII · p. 10

VIII. CONCLUSION

The study uses CFA to select five cognitively diverse models from a hybrid quantum-classical pool and fuse them. Across four binary classification benchmarks, the fusion outperformed the best individual models and recent state-of-the-art ensemble and quantum approaches.

VIII. CONCLUSION

  • The framework selects the top five most cognitively diverse models from a hybrid quantum-classical pool and combines them using score- and rank-based fusion.Strategies include uniform averaging and weighting by diversity strength and performance.Source: §VIII · p. 10
  • CFA-based fusion consistently outperformed the best individual models and recent state-of-the-art ensemble and quantum approaches across four binary classification benchmarks.The optimal combinations on all four datasets included both quantum and classical models.Source: §VIII · p. 10
  • The current pipeline is single-layer; evaluating whether fusion gains remain robust under quantum-induced noise is left for future work.The authors also propose expanding the pipeline into a Multi-layer Combinatorial Fusion framework.Source: §VIII · p. 10
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