QUACK: Quantum Aligned Centroid Kernel

TL;DR

The paper tackles the quadratic training cost of quantum kernel methods by introducing QUACK, a quantum kernel classifier that uses kernel alignment and centroid optimization to achieve linear training complexity while maintaining competitive accuracy relative to classical baselines. It encodes data with a trainable quantum embedding and optimizes centroid positions in data space, with inference scaling only with the number of centroids. In extensive simulations on eight datasets with up to 784 features, QUACK matches or closely approaches the performance of a classical SVM with an RBF kernel and demonstrates substantial reductions in circuit evaluations compared to full kernel methods. The work highlights QUACK’s potential as a scalable baseline for quantum-enhanced classification and outlines clear paths for hardware benchmarking and multi-class extensions.

Abstract

Quantum computing (QC) seems to show potential for application in machine learning (ML). In particular quantum kernel methods (QKM) exhibit promising properties for use in supervised ML tasks. However, a major disadvantage of kernel methods is their unfavorable quadratic scaling with the number of training samples. Together with the limits imposed by currently available quantum hardware (NISQ devices) with their low qubit coherence times, small number of qubits, and high error rates, the use of QC in ML at an industrially relevant scale is currently impossible. As a small step in improving the potential applications of QKMs, we introduce QUACK, a quantum kernel algorithm whose time complexity scales linear with the number of samples during training, and independent of the number of training samples in the inference stage. In the training process, only the kernel entries for the samples and the centers of the classes are calculated, i.e. the maximum shape of the kernel for n samples and c classes is (n, c). During training, the parameters of the quantum kernel and the positions of the centroids are optimized iteratively. In the inference stage, for every new sample the circuit is only evaluated for every centroid, i.e. c times. We show that the QUACK algorithm nevertheless provides satisfactory results and can perform at a similar level as classical kernel methods with quadratic scaling during training. In addition, our (simulated) algorithm is able to handle high-dimensional datasets such as MNIST with 784 features without any dimensionality reduction.

Figures (7)

• Figure 1: Architecture of the circuit if executed on hardware. The kernel entry $K_{ij}$ for samples $i$ and $j$ is the probability of measuring the all-zero bit string.
• Figure 2: Alternating optimization procedure for QUACK. The blue (green) dots show samples of class 1 (-1) in the embedding space $\Phi$. The superscript defines the current epoch and the subscript the dimension. The diamonds represent the centroids of the classes, where the superscript is the epoch in which the centroid has been optimized the last time and the subscript is the centroid class. For the KAO and CO steps, the superscript gives the epoch and the subscript the class for which the optimization is carried out with $+$ representing class 1 and $-$ representing class -1.
• Figure 3: Architecture of the encoding unitary U.
• Figure 4: Architecture of the unitary of a single parameterized layer $U_m(\boldsymbol{\theta}_m)$ with $n' = n - 1$.
• Figure 5: Test AUCs of the different models.