Hybrid Genetic Optimisation for Quantum Feature Map Design

TL;DR

The paper addresses automated quantum feature-map design for QSVM by substituting direct accuracy optimization with kernel-target alignment (KTA) as a fitness signal in NSGA-II genetic optimization, and by introducing a kernel-target alignment approximation to reduce kernel evaluations. It further proposes a hybrid approach where final trainable circuit parameters are refined with COBYLA after the genetic search. Across nine diverse binary classification datasets, the KTA-based methods achieve comparable test accuracy to the original approach while delivering larger training margins and reduced computational cost; the approximation offers additional speedups and a hybrid parameter training step often improves margins and sometimes accuracy. The work demonstrates that kernel-target alignment is a viable surrogate for accuracy in genetic design of quantum feature maps and that parametric training can complement structural optimization, with future directions including multi-phase genetic algorithms and alternative data-encoding strategies. $KTA = \frac{\langle K, O \rangle_F}{\sqrt{\langle K, K \rangle_F \langle O, O \rangle_F}}$ serves as the central metric, and the study highlights practical pathways for scalable quantum-feature-map design.

Abstract

Kernel methods are an important class of techniques in machine learning. To be effective, good feature maps are crucial for mapping non-linearly separable input data into a higher dimensional (feature) space, thus allowing the data to be linearly separable in feature space. Previous work has shown that quantum feature map design can be automated for a given dataset using NSGA-II, a genetic algorithm, while both minimizing circuit size and maximizing classification accuracy. However, the evaluation of the accuracy achieved by a candidate feature map is costly. In this work, we demonstrate the suitability of kernel-target alignment as a substitute for accuracy in genetic algorithm-based quantum feature map design. Kernel-target alignment is faster to evaluate than accuracy and doesn't require some data points to be reserved for its evaluation. To further accelerate the evaluation of genetic fitness, we provide a method to approximate kernel-target alignment. To improve kernel-target alignment and root mean squared error, the final trainable parameters of the generated circuits are further trained using COBYLA to determine whether a hybrid approach applying conventional circuit parameter training can easily complement the genetic structure optimization approach. A total of eight new approaches are compared to the original across nine varied binary classification problems from the UCI machine learning repository, showing that kernel-target alignment and its approximation produce feature map circuits enabling comparable accuracy to the previous work but with larger margins on training data (in excess of 20\% larger) that improve further with circuit parameter training.

Figures (39)

• Figure 1: An example illustrating how a feature map function could be used to make non-linearly separable points linearly separable in a higher dimensional space. In this case, the feature map could be implemented as a function that adds a third dimension to the points with decreasing value as distance from the central region of the points increases.
• Figure 2: A flow diagram outlining the algorithm followed to genetically train quantum feature map circuits. The diagram also shows how a hybrid method involving circuit parameter training can be performed after genetic optimization.
• Figure 3: The circuits with highest validation set accuracy produced by the three base genetic approaches when creating quantum feature maps for the Moons dataset. (a) shows the best produced circuit when training to maximize accuracy and minimize weighted size as in the original work, (b) shows the best circuit when training to maximize the exact kernel-target alignment and minimize unweighted size, and (c) shows the best circuit when training to maximize the approximation of the kernel-target alignment and minimize unweighted size. Circuits (b) and (c) are significantly larger. Unused gate layers and qubits are omitted from the diagrams.
• Figure 4: A graph showing the classification accuracies of the best models produced by various approaches of quantum feature map design on the Moons dataset, compared with a classical RBF kernel for reference. All approaches can be seen to achieve comparable accuracy across the different subsets.
• Figure 5: A graph showing the mean margin of the Moons training set points for the best classifiers produced by each approach, with errors bars showing standard deviation. Circuit parameter training and genetic training of kernel-target alignment are both shown to increase the mean margin size. The approach numbering corresponds to the numbering used in Figure \ref{['figure:moons_accuracies']}.