Quantum Generator Kernels

TL;DR

Quantum kernel methods promise enhanced data separability in high-dimensional quantum space, but practical QML on NISQ devices requires scalable data compression. We introduce Quantum Generator Kernels (QGKs), built from Variational Generator Groups (VGGs) that merge Lie-algebra generators into parameterized blocks, enabling data-dependent, expressive encodings with high parameter density per qubit. A linear projection pre-trains the mapping from input features to generator space via Kernel Target Alignment, producing a kernel computed as fidelity between quantum states. Empirical results across moons, circles, bank, MNIST, and CIFAR10 show QGKs outperform both state-of-the-art quantum kernels and classical baselines, while maintaining robustness to noise and hardware constraints; the framework offers a scalable, hardware-friendly path toward quantum-native kernel learning and a bridge to future fault-tolerant quantum execution.

Abstract

Quantum kernel methods offer significant theoretical benefits by rendering classically inseparable features separable in quantum space. Yet, the practical application of Quantum Machine Learning (QML), currently constrained by the limitations of Noisy Intermediate-Scale Quantum (NISQ) hardware, necessitates effective strategies to compress and embed large-scale real-world data like images into the constrained capacities of existing quantum devices or simulators. To this end, we propose Quantum Generator Kernels (QGKs), a generator-based approach to quantum kernels, comprising a set of Variational Generator Groups (VGGs) that merge universal generators into a parameterizable operator, ensuring scalable coverage of the available quantum space. Thereby, we address shortcomings of current leading strategies employing hybrid architectures, which might prevent exploiting quantum computing's full potential due to fixed intermediate embedding processes. To optimize the kernel alignment to the target domain, we train a weight vector to parameterize the projection of the VGGs in the current data context. Our empirical results demonstrate superior projection and classification capabilities of the QGK compared to state-of-the-art quantum and classical kernel approaches and show its potential to serve as a versatile framework for various QML applications.

Figures (4)

• Figure 1: Quantum Generator Kernel: A generator-based quantum kernel architecture based on VGGs for parameterizable projection. Each colored matrix corresponds to one of $g=93$Variational Generator Groups (VGGs) merged for $\eta=5$ qubits, visualized as heatmaps of the magnitude (blue) and phase (green) of the resulting generators merged into the operator. The QGK is parameterized by the context $\bm{\phi}$, which is either given directly by the input or extracted from the input using a feature extractor, adapted during the pre-training phase (1) by updating the parameters ${\bm{\theta}}$ to minimize the Kernel-Target Alignment (KTA) loss. In phase (2), a support vector machine (SVM), parameterized by $\bm{\alpha}$, is trained using the resulting QGK $\hat{{\bm{U}}}$.
• Figure 2: Comparing the properties of the QGK to classical and quantum kernels w.r.t. the number of available qubits $\eta$, regarding the: \ref{['fig:scale:params']} number of available parameters, \ref{['fig:scale:entangle']} entanglement capability by means of the Meyer-Wallach measure, \ref{['fig:scale:expressibility']} expressibility by means of the spectral concentration, and \ref{['fig:scale:complexity']} computational complexity, showing the complexity breakeven dataset size $n$ when classically simulating the QGK (left y-axis) and the number of VGGs and resulting inputs (right y-axis, blue), scaled to $n=10d$ ($n\gg d$). The properties of the QEK$^*$, HEE$^*$, and PQK$^*$ are reported w.r.t the input scalability shown in \ref{['fig:scale:params']}. Refer to \ref{['app:grouping_hp']} for an extended analsyis.
• Figure 3: Training performance of the QGK(blue), QEK(red), PQK(dark green), HEE(green), RBF(yellow), and Linear(orange) Kernels and MLP(grey) w.r.t. the Test Accuracy (top) and KTA (bottom) in the moons, circles, bank, MNIST, and CIFAR10 benchmarks, with the QGK outperforming all compared quantum kernels and classical kernels baselines. Final accuracies are summarized in \ref{['tab:final_acc']}.
• Figure 4: VGG grouping analysis and comparison regarding \ref{['fig:app:entanglement']} the entanglement capability by means of the Meyer-Wallach measure and \ref{['fig:app:expressibility']} the expressibility by means of the spectral concentration, accross four input scalings: linear ($g=\eta$, left) yielding the most generators per group, quadratic ($g=\eta^2$), exponential ($g$ according to Eqs. \ref{['eq:groups']} and \ref{['eq:generators_per_group']}), and all (right) using the maximum available input dimensions, i.e., the total number of generators $4^\eta-1$ and three projection widths: the default $w=\eta$ causing a wide stride, i.e., assignment of distant generators to groups, $w=1$, causing a medium stride, and $w=0$, causing a narrow stride. \ref{['fig:app:groupsize_exp']} shows a comparison of spectral concentration across group sizes $g\in[15,60,240]$, averaged over projection widths ($w \in \{0,\,1,\,\eta\}$), with error bars indicating the standard deviation. \ref{['fig:projection comparison']} shows projection heatmaps for VGG$_{0,1,2,19,20}$ for $\eta=3$ qubits, with widths from $w=0$ (upper) to $w=\eta$ (lower), visualizing the magnitude (blue) and phase (green) of the resulting generator matrices $\hat{{\bm{H}}}$.

Theorems & Definitions (15)

• Theorem 3.1
• proof
• Proposition D.1: Expressive Robustness under Generator Grouping
• Definition D.2: Robust Grouping Criteria
• Proposition D.3: Sufficient Condition for Linear Independence
• Theorem D.4: Expressibility Bounds for Grouped Generators
• proof : Proof sketch.
• Proposition E.1: RKHS-Constrained Expressivity
• Lemma F.2
• Lemma F.3: Simplify complexity bounds