Quantum Hamiltonian Embedding of Images for Data Reuploading Classifiers

TL;DR

This work investigates quantum machine learning for image classification by challenging the primacy of runtime speedups and instead leveraging classical deep learning heuristics. It introduces a quantum classifier that combines Hamiltonian data embedding with a data reuploading circuit, using $W(t;M)=e^{-i H_M t/2}$ with $H_M=(M+M^T)/2$ to encode images and a repeated $L$-layer structure $\prod_{i=1}^L [V(\omega_i) W(t_i;M)] |+\rangle^{\otimes n}$ for processing. Numerically, the HamEmb+DataReUploading model outperforms the baseline QCNN across MNIST, FashionMNIST, and related subsets, achieving up to ~40% improvements on MNIST test accuracy, and the authors extract six design principles to guide future QML model development. The results underscore the value of integrating inductive biases and DL-inspired heuristics into quantum models, potentially reducing reliance on heavy classical preprocessing and informing practical QML hardware implementations.

Abstract

When applying quantum computing to machine learning tasks, one of the first considerations is the design of the quantum machine learning model itself. Conventionally, the design of quantum machine learning algorithms relies on the ``quantisation" of classical learning algorithms, such as using quantum linear algebra to implement important subroutines of classical algorithms, if not the entire algorithm, seeking to achieve quantum advantage through possible run-time accelerations brought by quantum computing. However, recent research has started questioning whether quantum advantage via speedup is the right goal for quantum machine learning [1]. Research also has been undertaken to exploit properties that are unique to quantum systems, such as quantum contextuality, to better design quantum machine learning models [2]. In this paper, we take an alternative approach by incorporating the heuristics and empirical evidences from the design of classical deep learning algorithms to the design of quantum neural networks. We first construct a model based on the data reuploading circuit [3] with the quantum Hamiltonian data embedding unitary [4]. Through numerical experiments on images datasets, including the famous MNIST and FashionMNIST datasets, we demonstrate that our model outperforms the quantum convolutional neural network (QCNN)[5] by a large margin (up to over 40% on MNIST test set). Based on the model design process and numerical results, we then laid out six principles for designing quantum machine learning models, especially quantum neural networks.

Figures (11)

• Figure 1: The quantum machine learning model described in Eqn. \ref{['eqn:actual-model']}. The model shown in the figure has a five-qubit circuit for the images from the FashionMNIST dataset (and MNIST as well). The grey-scale images are padded from $28\times 28$ to $32\times 32$ with zeros. Then the quantum Hamiltonian $H_M$ is constructed with the padded image matrices. The data encoding unitary (grey box in the circuit diagram) and the parameterised circuit unitary (white box in the circuit diagram) are repeated $L$ times for an $L$-layered data re-uploading circuit.
• Figure 2: Different layouts for the SU(4) gate in three- and five-qubit circuits.
• Figure 3: Sample images from the Kaggle CT Medical Image dataset. \ref{['fig:ctimgoriginal']}: Original image samples from the CT medical image dataset. \ref{['fig:ctimgresized']}: Same images as Fig. \ref{['fig:ctimgoriginal']}, but resized to 32 by 32. The resize is achieved via OpenCV-Python's resize function. The pixels are also normalised before being fed into the quantum neural networks using OpenCV-Python's normalize function.
• Figure 4: Sample images from the Sklearn digits dataset. The size of the images is eight by eight.
• Figure 5: Sample images from the MNIST dataset. The size of the original images is 28 by 28. Images were padded with zeros to 32 by 32 before constructing the Hermitian operators of the images.