Quantum Support Vector Classifiers Overview

• Quantum Support Vector Classifiers (QSVCs) are quantum algorithms that compute kernels by mapping classical data into high-dimensional Hilbert spaces via parameterized quantum circuits.
• They employ shallow single-qubit rotations and projective measurements to evaluate quantum kernel functions, offering noise resilience and competitive performance compared to classical SVMs.
• QSVCs enable efficient supervised classification for moderate datasets by balancing the trade-offs between circuit depth, number of required measurements, and scalability on NISQ hardware.

Quantum Support Vector Classifiers (QSVCs) are quantum algorithms that generalize classical support vector machines (SVMs) by computing the kernel matrix in a high-dimensional Hilbert space using quantum feature maps. By leveraging quantum circuits to encode classical data, QSVCs aim to enhance expressivity and computational efficiency for supervised classification tasks. The core construct is a quantum kernel function, defined as the squared overlap of quantum states associated with classical input vectors, evaluated via projective measurements after applying parameterized quantum circuits. QSVCs have been empirically demonstrated to match or sometimes exceed the performance of classical SVMs, while exhibiting distinct scaling, robustness, and resource profiles. This article provides a comprehensive overview of QSVC methodologies, circuit constructions, training protocols, comparative performance, scaling laws, and hardware considerations.

1. Quantum Kernel Construction and Feature Maps

QSVCs replace the classical kernel function with a quantum-computed kernel, typically

$$k(x_i, x_j) = |\langle 0^{\otimes n} | U_\phi(x_j)^\dagger U_\phi(x_i) | 0^{\otimes n} \rangle |^2,$$

where $x_i, x_j \in \mathbb{R}^d$ are classical feature vectors, $n = d$ is the number of qubits, and $U_\phi(x)$ is a parameterized quantum circuit encoding $x$.

A canonical QSVC feature map consists of two sequential blocks of single-qubit rotations:

• Encoding Block: $U_\phi(x) = \bigotimes_{j=1}^n R_Y(x'_j)$ with $R_Y(\theta) = \exp(-i\theta \sigma_Y/2)$ and $x'_j$ rescaled to $[0,\pi]$.
• Kernel Evaluation: For input pairs $(x_i, x_j)$, prepare $|0\rangle^{\otimes n}$, apply $U_\phi(x_i)$, then $U_\phi(x_j)^\dagger$, and measure the probability of the all-zeros state to estimate the kernel (Pinheiro et al., 12 Sep 2025).

No ancillary qubits or complex two-qubit gates are required for this kernel evaluation; the depth is solely determined by the number of single-qubit rotations, ensuring NISQ-compatibility. Generalizations include alternative feature maps (ZFeatureMap, ZZFeatureMap, PauliFeatureMap), entangling operations, and optimizable parameter layers, as evidenced in different software implementations (Tasar et al., 2023, Villalba-Ferreiro et al., 1 Dec 2025).

2. Training Procedure and Multi-Class Protocols

Training a QSVC proceeds identically to a classical kernel SVM once the quantum kernel matrix $K_{ij}$ is assembled:

1. Kernel Matrix Estimation: For $N$ training samples, compute all $N^2$ pairwise kernel entries by quantum circuit evaluation and projective measurement.
2. Classical Quadratic Programming: Given $K$ and binary labels $y_i \in \{+1,-1\}$, solve the QP

$$\underset{\alpha}{\operatorname{minimize}} \quad \frac{1}{2} \alpha^T (YKY) \alpha - \mathbf{1}^T \alpha$$

subject to $0 \leq \alpha_i \leq C$ and $\sum_i \alpha_i y_i=0$, where $Y = \operatorname{diag}(y_i)$.

1. Multi-class Extensions: Both one-vs-rest (OvR) and hierarchical two-step decompositions are supported. In OvR, $C$ binary problems are trained in parallel; in hierarchical, one class is separated first, then a binary classifier discriminates among the remainder (Pinheiro et al., 12 Sep 2025).

These protocols are compatible with any classical SVM solver; typically, scikit-learn's SVC is used with class-weighted penalties to compensate for label imbalance (Pinheiro et al., 12 Sep 2025). Predictions for new points require $O(N)$ kernel function evaluations per test sample.

3. Computational Complexity, Scaling, and Hardware Requirements

The resource scaling of quantum kernel-based QSVCs is governed by:

• Gate Count: Each kernel evaluation involves $O(d)$ single-qubit gates; total quantum gate count for training is $O(N^2 d)$.
• Prediction Cost: Predicting on $M$ new instances requires $O(MNd)$ quantum circuit executions.
• Measurement Overhead: Each entry in $K$ requires multiple measurement shots for statistical confidence. For $N=300$, $d=10$ (SDSS example), about $1.8\times10^6$ rotations are needed per training round, exclusive of measurement repetition (Pinheiro et al., 12 Sep 2025).

By contrast, quantum least-squares SVMs using the HHL algorithm exhibit constant-in-$N$ circuit depth once data is pre-processed to a reduced set of $n\ll N$ representative elements: time scales as $\operatorname{poly}(\log n, \kappa, 1/\epsilon)$, where $\kappa$ is the condition number and $\epsilon$ the inversion accuracy. However, HHL-based circuits are deeper—requiring quantum phase estimation, controlled rotations, and uncomputation steps—and are highly susceptible to noise (Pinheiro et al., 12 Sep 2025).

This scaling distinction creates a trade-off: QSVC is preferred for moderate $N$ and $d$, while HHL-based approaches are attractive for massive datasets given fault-tolerant hardware and robust circuit decompositions.

4. Empirical Performance and Comparative Evaluation

The empirical efficacy of QSVCs centers on their ability to closely match, and sometimes marginally exceed, classical SVMs in supervised classification benchmarks. On the reduced SDSS dataset ($d=10$, $N=300$), the following summary statistics were obtained for the two-step multiclass scheme (Pinheiro et al., 12 Sep 2025):

Model	Accuracy	F1-Score
QSVC (quantum kernel)	$0.969 \pm 0.003$	$0.950 \pm 0.006$
Classical SVM	$0.968 \pm 0.001$	$0.950 \pm 0.004$
HHL LS-QSVM	$0.893$	$0.812$
HHL LS-CSVM	$0.914$	$0.872$

HHL-based QSVCs exhibit particularly degraded performance on minority-class (QSO) isolation tasks and under hardware-induced noise, while remaining competitive for majority-class separation. QSVCs retain shallow circuits and thus greater resilience under decoherence, with the main bottleneck being the number of circuit executions rather than depth.

Generalization to other datasets (e.g., Iris, MNIST, and high-dimensional astrophysics and finance datasets) has consistently shown quantum kernels to be at least as performant as best-in-class classical kernels, provided feature maps are suitable (Tasar et al., 2023, Villalba-Ferreiro et al., 1 Dec 2025, Chen et al., 2024, Bhattacharjee et al., 2024).

5. Hardware Suitability, Noise, and NISQ Era Considerations

QSVCs are explicitly designed to be executable on near-term quantum devices:

• Circuit Depth: Only two layers of parameterized single-qubit rotations are required per kernel evaluation; no deep entangling structures or mid-circuit measurements are involved (Pinheiro et al., 12 Sep 2025).
• Noise Sensitivity: The shallow depth and absence of complex gate sequences increase robustness to decoherence. The main limitation is the requirement for repeated measurements to statistically estimate kernel entries (Pinheiro et al., 12 Sep 2025).
• NISQ Feasibility: On current hardware, the large number of required shots for kernel estimation and quadratic scaling in dataset size are prohibitive for large-scale deployment, but practical for hundreds to low thousands of samples and features.

By contrast, HHL-based QSVCs, while asymptotically more efficient in data size, mandate much deeper circuits, extensive phase estimation, and are experimentally observed to degrade significantly under current device noise, with poor classification outcomes (Pinheiro et al., 12 Sep 2025).

6. Extensions: Universality, Circuit Design, and Algorithmic Variants

QSVCs are universally expressive in principle: It has been proven that suitably constructed quantum feature maps (e.g., those derived from the $k$-Forrelation problem) can render the associated kernel PromiseBQP-complete, thereby allowing QSVCs to efficiently classify any problem in BQP given polynomial resources (Jäger et al., 2022). This universality extends to variational quantum classifiers (VQCs) with trainable post-feature-map circuits and to automatic quantum circuit synthesis of data-driven feature maps via multiobjective evolutionary algorithms (Altares-López et al., 2021).

Moreover, the separation between quantum and classical SVMs can in principle be mapped to the intractability of simulating high-dimensional quantum Hilbert space embeddings classically. However, in practical low- to medium-dimensional settings, current quantum and classical SVM performance remains closely matched (Pinheiro et al., 12 Sep 2025, Tasar et al., 2023).

7. When to Prefer Quantum Kernel QSVCs

The preferred operational regimes for kernel-based QSVCs are:

• Moderate-size datasets ($N\lesssim 10^3$, features up to a few tens), where O($N^2 d$) circuit executions are manageable and quantum hardware can exploit the shallowness of angle-encoding circuits.
• Noise-resilient deployments on NISQ devices due to the low circuit depth and modest resource requirements.
• Situations where explicit quantum feature maps grant access to high expressivity or task-specific kernels that are challenging for classical algorithms.

For extremely large datasets ($N\gg 10^4$), or where circuit depth is not the limiting factor but count of executions is, HHL-based LS-SVMs or classical SVMs often remain more practical until scalable, fault-tolerant quantum memory and more efficient kernel-estimation strategies become available (Pinheiro et al., 12 Sep 2025).

---

Key references:

• Pinheiro et al., “Quantum kernel and HHL-based support vector machines for multi-class classification” (Pinheiro et al., 12 Sep 2025)
• Taşar et al., “Machine Learning in the Quantum Age: Quantum vs. Classical Support Vector Machines” (Tasar et al., 2023)
• Jäger & Krems, “Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines” (Jäger et al., 2022)
• Deng et al., “Validating Large-Scale Quantum Machine Learning: Efficient Simulation of Quantum Support Vector Machines Using Tensor Networks” (Chen et al., 2024)
• Ezawa, “Variational Quantum Support Vector Machine based on $\Gamma$ matrix expansion and Variational Universal-Quantum-State Generator” (Ezawa, 2021)
• Nadim et al., “Performance Analysis of Quantum Support Vector Classifiers and Quantum Neural Networks” (Villalba-Ferreiro et al., 1 Dec 2025)