Four-Class Classifier Overview

• Four-class classification is a system that assigns inputs to one of four exclusive classes using decision rules based on maximum probability or raw logits.
• Quantum implementations like QCNN and SWAP-Test leverage amplitude encoding, parameter-shift training, and specialized measurement mapping to achieve high accuracy.
• Classical methods such as OSELM and CASIMAC use multi-neuron outputs and simplex-based kernel techniques to deliver robust, calibrated performance.

A four-class classifier is a classification algorithm designed to assign each input sample to one of exactly four mutually exclusive classes. This paradigm generalizes the binary classification scenario to a multiclass setting and is relevant across quantum and classical machine learning, signal processing, and information retrieval. Four-class classification exposes algorithmic and representational considerations distinct from either binary or general $N$-class scenarios, enabling both focused benchmarking and technical analysis of quantum and classical classifier architectures.

1. Formal Definition and Decision Rule

Let $X$ denote the input feature space and $Y = \{1,2,3,4\}$ denote the set of possible labels. A four-class classifier is a function $f: X \to Y$ constructed so that for any $x \in X$, $f(x)$ assigns a single label $y \in Y$ corresponding to the predicted class for $x$. In probabilistic settings, the classifier may additionally output a vector of class probabilities $p = [p_1, p_2, p_3, p_4]$ with $\sum p_i = 1$, and prediction is performed via $f(x) = \arg\max_{i=1}^4 p_i$.

The key performance metrics are classification accuracy, cross-entropy loss, and in some settings, calibration error (e.g., ECE for calibrated confidence estimation) (Heese et al., 2021). The prediction rule is thus

$\hat{y} = \arg\max_{i=1,2,3,4} s_i(x)$

where $s_i(x)$ is either a raw logit or a probability for each class.

2. Quantum Classifier Architectures for Four Classes

Quantum machine learning has produced multiple families of four-class classifiers tailored to NISQ-era hardware, notably the Quantum Convolutional Neural Network (QCNN) and the SWAP-Test Classifier.

Quantum Convolutional Neural Networks (QCNN)

In "Multiclass classification using quantum convolutional neural networks with hybrid quantum-classical learning" (Bokhan et al., 2022) and "Multi-Class Quantum Convolutional Neural Networks" (Mordacci et al., 2024), QCNNs for four classes process images as follows:

• Data Encoding: Amplitude encoding of a normalized vectorized image, often preprocessed by PCA to fit $2^n$ amplitudes into $n$ qubits (e.g., 256 features $\to$ 8 qubits).
• Circuit Architecture: Layered quantum circuits with convolutional (single- and multi-qubit gates) and pooling (qubit reduction via entangling gates or tracing out) stages, ultimately mapping data qubits to a set of two or more measured qubits.
• Measurement and Output: Final measured qubits yield a four-dimensional readout, either one-hot via ancillas (Bokhan et al., 2022) or by mapping computational basis states $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ to the four classes (Mordacci et al., 2024).
• Training Objective: Classical softmax and cross-entropy loss over output probabilities, with parameters updated by the parameter-shift rule and optimizers such as Adam.
• Numerical Results: On PCA-reduced MNIST (digits $\{0,1,2,3\}$), QCNNs achieve test accuracies in the $85$--$93\%$ range, narrowly trailing compact classical CNN baselines ($90$--$97\%$) under matched parameter budgets (Bokhan et al., 2022, Mordacci et al., 2024).

SWAP-Test Based Multi-Class Classifiers

The Multi-Class SWAP-Test classifier (Pillay et al., 2023) applies the following scheme:

• Data and Label-State Encoding: Each sample is encoded as a quantum state via a feature map $U_{\Phi}(x)$; each class $i$ gets a fixed single-qubit "label state" $|\ell_i\rangle$ whose Bloch vector $y_i$ is placed as one vertex of the Tammes-optimal tetrahedron on the Bloch sphere.
• Circuit Topology: Utilizes a modified SWAP-test on two data registers, a label qubit, an ancilla, and an index register for training samples.
• Measurement and Assignment: Single-qubit tomography on the label register yields a vector $$y_{\mathrm{pred}} = \sum_{i=1}^4 \alpha_i y_i; \ \alpha_i$$ are weights computed from training overlaps.
• Decision Rule: Assign the class index maximizing the inner product $y_i \cdot y_{\mathrm{pred}}$.
• Noise Robustness: The scheme is invariant to depolarizing noise on the label qubit, as the predicted vector uniformly shrinks but angular relations are preserved unless label-vectors become non-separable.
• Empirical Results: On 4-XOR synthetic data, ideal simulations yield $100\%$ accuracy; finite-sampling and depolarizing noise up to $p=0.1$ do not degrade accuracy (Pillay et al., 2023).

Polyadic Quantum Classifier

The Polyadic Quantum Classifier (Cappelletti et al., 2020) supports $2^N$-class prediction with $N$ measured qubits. For four-class problems:

• Circuit: 2 entangled qubits with repeated data encoding, entangling, and trainable rotation blocks.
• Output Mapping: Measurement in the computational basis; bitstrings mapped to class indices (00 $\rightarrow$ class 0, ..., 11 $\rightarrow$ class 3).
• Loss and Optimization: Mini-batch cross-entropy loss, parameter-shift and gradient-free optimizers.
• Numerical Results: On a 2D four-Gaussian benchmark, achieves $85\% \pm 2.8\%$ accuracy (simulated QPU), compared to XGBoost at $88\%$ (Cappelletti et al., 2020).

3. Classical Four-Class Classifier Methodologies

Four-class classification in classical learning typically leverages well-established architectures with minimal adaptation:

• Online Universal Classifier (OSELM): An online sequential Extreme Learning Machine with $L=4$ output neurons (one-hot encoding). The only modified hyperparameters relative to binary or three-class settings are output dimension and initialization block size (Er et al., 2016). In this regime, each prediction is handled as $\hat{j} = \arg\max_{j=1,\ldots,4} \hat{y}_j$ where $\hat{y} = \beta^T h(x)$.
• Expected Accuracy: For moderate-scale four-class problems, OSELM achieves $80$--$95\%$ accuracy, with millisecond-scale online updates and sub-millisecond prediction times (Er et al., 2016).
• Calibrated Simplex-Mapping Classifier (CASIMAC): Embeds the four labels as vertices of a regular tetrahedron ($\mathbb{R}^3$ simplex), mapping data to the corresponding simplex region via k-nearest-neighbor attraction/repulsion and kernel ridge or Gaussian process regression. Predictions are given by the closest vertex in latent space; confidence is provided by Monte Carlo estimation of the Gaussian probability mass within each class-region (Heese et al., 2021).

4. Comparative Table of Representative Four-Class Classifiers

Below is a summary of classifier type, quantum/classical regime, and characteristic features (restricted to explicit content in the provided data).

Classifier	Regime	Output/Decision Scheme
QCNN (Bokhan et al., 2022Mordacci et al., 2024)	Quantum NISQ	Ancilla or basis state → softmax 4-probabilities
SWAP-Test (Pillay et al., 2023)	Quantum NISQ	Label state tomography → vector inner product
Polyadic (Cappelletti et al., 2020)	Quantum NISQ	$N$-qubit bitstring mapped to class index
OSELM (Er et al., 2016)	Classical online	4 output neurons, $\arg\max_j$ decision
CASIMAC (Heese et al., 2021)	Classical, kernel	Closest simplex vertex in $\mathbb{R}^3$

5. Optimization, Training, and Theoretical Properties

• Optimization in Quantum Models: Parameter-shift rule for gradient estimation dominates, leveraging two-point evaluations for efficient, hardware-compatible training (Bokhan et al., 2022, Mordacci et al., 2024, Cappelletti et al., 2020). Adam is standard for classical parameter updates, with learning rates in $[5\times10^{-5},\,10^{-2}]$ and minibatch training.
• Classical Algorithms: Recursive least squares or direct kernel ridge/GPR fitting; no explicit learning rate (Er et al., 2016, Heese et al., 2021).
• Bayes-Optimality: In the quantum detection-theory framework, the four-class classifier is realized by optimizing a set of four projective or POVM measurements $\{\Pi_k\}$ minimizing the average error $R$ (Helstrom bound). This yields a classifier reducing to classical one-vs-rest in the commuting case but admits strictly lower average risk with quantum-encoded or entangled class/states (Tiwari et al., 2018).

6. Performance, Robustness, and Scalability

• Four-Class QCNNs: Achieve $85$--$93\%$ accuracy on subsets of MNIST (digits $\{0,1,2,3\}$, $\{3,4,5,6\}$), with network sizes $<$200 trainable parameters and training regimes of 10--50 epochs (Bokhan et al., 2022, Mordacci et al., 2024).
• SWAP-Test Classifier: Yields $99$--$100\%$ accuracy on ideal and noise-limited synthetic 4-class problems; is robust to depolarizing noise due to invariant angular assignment (Pillay et al., 2023).
• Polyadic Quantum Classifier: Attains $85\% \pm 2.8\%$ on synthetic four-class data, with two-qubit circuits and no need for deep variational circuits (Cappelletti et al., 2020).
• Classical Baselines: OSELM achieves $80$--$95\%$ across several real-world problems; CASIMAC matches kernel SVM and GPC accuracy while additionally providing calibrated confidence estimates (Er et al., 2016, Heese et al., 2021).

7. Technical and Practical Considerations

• Encoding Overhead: Amplitude encoding of high-dimensional classical features into $n$ qubits is bottlenecked by $O(2^n)$ circuit complexity; PCA and dimensionality reduction are typically used to fit classical data into tractable numbers of qubits for quantum classifiers (Bokhan et al., 2022, Mordacci et al., 2024).
• Readout and Calibration: Direct mapping of measurement outcomes to class indices ensures hardware efficiency in both Polyadic and QCNN architectures. Methods such as CASIMAC provide explicit statistical calibration via simplex-based geometries and kernel regression (Heese et al., 2021).
• NISQ Suitability: Quantum architectures (QCNN, Polyadic, SWAP-Test) are explicitly designed to constrain circuit depth and qubit count, employing repeated pooling, sparse multi-qubit gates, and data-reuploading (Bokhan et al., 2022, Mordacci et al., 2024, Cappelletti et al., 2020, Pillay et al., 2023).
• Scalability: For all NISQ quantum schemes, increasing the class number $L$ typically requires either increased qubit count ($n = \lceil\log_2 L\rceil$) or expanded measurement schemes; circuit compression and parameter-sharing techniques are discussed as possible improvements (Bokhan et al., 2022, Cappelletti et al., 2020).

• Multiclass classification using quantum convolutional neural networks with hybrid quantum-classical learning (Bokhan et al., 2022)
• Multi-Class Quantum Convolutional Neural Networks (Mordacci et al., 2024)
• Polyadic Quantum Classifier (Cappelletti et al., 2020)
• A Multi-Class SWAP-Test Classifier (Pillay et al., 2023)
• An Online Universal Classifier for Binary, Multi-class and Multi-label Classification (Er et al., 2016)
• Calibrated simplex-mapping classification (Heese et al., 2021)
• Multi-class Classification Model Inspired by Quantum Detection Theory (Tiwari et al., 2018)