Efficient Parameter Optimisation for Quantum Kernel Alignment: A Sub-sampling Approach in Variational Training

TL;DR

This work applies the sub-sampling method to synthetic datasets and a real-world breast cancer dataset and demonstrates considerable reductions in the number of circuits required to train the quantum kernel while maintaining classification accuracy.

Abstract

Quantum machine learning with quantum kernels for classification problems is a growing area of research. Recently, quantum kernel alignment techniques that parameterise the kernel have been developed, allowing the kernel to be trained and therefore aligned with a specific dataset. While quantum kernel alignment is a promising technique, it has been hampered by considerable training costs because the full kernel matrix must be constructed at every training iteration. Addressing this challenge, we introduce a novel method that seeks to balance efficiency and performance. We present a sub-sampling training approach that uses a subset of the kernel matrix at each training step, thereby reducing the overall computational cost of the training. In this work, we apply the sub-sampling method to synthetic datasets and a real-world breast cancer dataset and demonstrate considerable reductions in the number of circuits required to train the quantum kernel while maintaining classification accuracy.

Figures (11)

• Figure 1: A) Quantum kernel alignment uses n qubits to encode the dataset into a feature map, estimate the kernel, and optimize parameters $\boldsymbol{\theta}$ until kernel separation improves. B) In the sub-sampling method, subsets with $k$ elements of the dataset are similarly processed using n qubits. This cycle repeats with different subsets, optimizing $\boldsymbol{\theta}$ until the entire dataset is sampled, yielding an optimized kernel. This will end with the estimation of the full dataset, optimised kernel.
• Figure 2: Total number of queries and speed-up relative to full kernel for the top 12 results using the SPSA optimizer, grouped by Hardware Efficient (HE) ansatze and Real Amplitudes (RA) ansatze. Upper panels: the range of ROC AUC values is 0.975-0.990 and the F1 scores range from 0.856-0.931 on the ZZ dataset (see Table \ref{['apx-spsa-zz']} for full results). Lower panels: all ROC AUC values are 1.0 and the F1 values range from 0.969-1.0 on the Coset dataset (see Table \ref{['apx-spsa-coset']} for full results). In both panels, $k$ is the sub-kernel size, and $s$ is the number of samples. Results are ordered by descending $k$.
• Figure 3: Combined GNN-extracted pati2022hierarchical hierarchical cellular features with our sub-sampling method to optimise the creation of a quantum kernel used to classify subtypes of cancer. The GNN-extracted embeddings are used to create the dataset that will be sampled using the methodology discussed in this work. The output of the workflow is a fully variationally optimised quantum kernel.
• Figure 4: AF vs D
• Figure 5: NBU vs AFD