Scalable Tensor Network Simulation for Quantum-Classical Dual Kernel

TL;DR

The study tackles the instability of quantum-kernel SVMs at scale by coupling a quantum kernel, computed via tensor-network contraction of a block-product-state circuit, with a classical RBF kernel in a learnable dual-kernel $K_{q-c}=\alpha K_q+(1-\alpha)K_c$. It demonstrates scalable evaluation up to $n\le 784$ and shows that the dual kernel outperforms pure quantum and classical baselines on Fashion-MNIST tasks, particularly by maintaining generalization as dimensionality grows. The results highlight the classical component as a stabilizing anchor against exponential concentration and hardware noise, while preserving quantum gains at moderate dimensions, and quantify the benefits through cross-validated mixing weights. The work provides a practical pathway toward robust, large-scale quantum kernel learning with low overhead and broad applicability, and points to future directions in adaptive kernel fusion and explicit noise modeling.

Abstract

This paper presents an efficient and scalable tensor network framework for quantum kernel circuit simulation, alleviating practical costs associated with increasing qubit counts and data size. The framework enables systematic large-scale evaluation of a linearly mixed quantum-classical dual kernel of up to 784 qubits. Using Fashion-MNIST, the classification performance of the test dataset is compared between a classical kernel, a quantum kernel, and the quantum-classical dual kernel across the feature dimensions from 2 to 784, with a one-to-one mapping between encoded features and qubits. Our result shows that the quantum-classical dual kernel consistently outperforms both single-kernel baselines, remains stable as the dimensionality increases, and mitigates the large-scale degradation observed in the quantum kernel. Analysis of the learned mixing weights indicates that quantum contributions dominate below 128 features, while classical contributions become increasingly important beyond 128, suggesting that the classical kernel provides a stabilizing anchor against concentration effects and hardware noise while preserving quantum gains at lower dimensions.

Figures (5)

• Figure 1: QSVM framework and its circuit realization. (A) Classical data vectors are mapped to quantum states via parameterized circuits $U(x_i)$, whose overlaps are obtained through $U(x_i)U^{\dagger}(x_j)$. (B) Tensor-network–representable structure of the QSVM with the specific parameterized quantum circuit used in this work. Each data is presented as angular information of the quantum circuit.
• Figure 2: Framework of the quantum–classical dual-kernel SVM. Data are first preprocessed via standard scaling, PCA, and min–max scaling, followed by parallel kernel construction using QSVM with tensor network approach and classical RBF SVM. Then the quantum $K_q$ and classical $K_c$ kernels are linearly combined to form the hybrid kernel $K_{q-c}$ which is used for final SVM classification. Note that all hyperparameter (symbols with $*$) are selected by cross-validation.
• Figure 3: Accuracy as a function of the number of features/qubits for classical kernel, quantum kernel, and the quantum-classical dual-kernel across 45 binary classification tasks on the Fashion-MNIST dataset. (A) Training-set performance shows that all models achieve increasing accuracy with higher feature dimensionality. (B)Testing-set performance initially improves with increasing model complexity but degrades at large $n$ due to overfitting, which is particularly severe for the quantum kernel model. Across all qubit regimes, the quantum-classical dual-kernel consistently achieves higher average accuracy, demonstrating its robustness as a hybrid optimization approach.
• Figure 4: Confusion matrices for 10-class Fashion-MNIST classification at $n=64$ comparing SVM, QSVM, and dual-kernel SVM. The classical and dual-kernel SVMs exhibit comparable classification performance and both outperform the pure quantum kernel model.
• Figure 5: Average testing accuracy of the dual-kernel SVM as a function of the weighting coefficient $\alpha$ at $n=64$ over 45 binary tasks. The accuracy curve shows optimal performance at intermediate $\alpha$ values and degradation toward the pure quantum-kernel regime.