Neural Quantum Embedding: Pushing the Limits of Quantum Supervised Learning

TL;DR

This work introduces Neural Quantum Embedding (NQE), a framework that uses a classical neural network to learn data-dependent quantum embeddings, thereby maximizing the trace distance between class-conditional quantum states and tightening the empirical risk lower bound. By replacing or augmenting fixed trainable unitary embeddings with NQE, the approach achieves higher data separability, improved training and generalization, and increased robustness to noise across quantum neural networks and quantum kernel methods. Empirical results on MNIST-based tasks with four-qubit QCNNs and IBM hardware show substantial accuracy gains (up to 96% with PCA-NQE) and reduced generalization bounds, with consistent improvements across larger quantum systems and Fashion-MNIST in simulations. The findings indicate that constraining expressibility to enhance separability enables more trainable and scalable quantum learning on NISQ devices, offering practical benefits for quantum-assisted classification.

Abstract

Quantum embedding is a fundamental prerequisite for applying quantum machine learning techniques to classical data, and has substantial impacts on performance outcomes. In this study, we present Neural Quantum Embedding (NQE), a method that efficiently optimizes quantum embedding beyond the limitations of positive and trace-preserving maps by leveraging classical deep learning techniques. NQE enhances the lower bound of the empirical risk, leading to substantial improvements in classification performance. Moreover, NQE improves robustness against noise. To validate the effectiveness of NQE, we conduct experiments on IBM quantum devices for image data classification, resulting in a remarkable accuracy enhancement from 0.52 to 0.96. In addition, numerical analyses highlight that NQE simultaneously improves the trainability and generalization performance of quantum neural networks, as well as of the quantum kernel method.

Figures (13)

• Figure 1: Overview of the NQE training. The unitary transformation that maps $x_i$ to the quantum feature space is determined by the output of a classical neural network denoted by $g(x_i,w)$, where $w$ represents trainable parameters. The resulting quantum state is $|x_i(w)\rangle = V(g(x_i,w))|0\rangle^{\otimes n}$. The goal of the training is to produce mapping functions that can separate the two classes of data into two orthogonal subspaces. Efficient calculation of the fidelity between the two quantum states produced by the feature map is performed using a quantum computer.
• Figure 2: (a) Schematic representation of the quantum circuit used in the experiments. The green rectangle indicates the Neural Quantum Embedding (NQE), which transforms classical data $x_i$ into a quantum state $\vert x_i \rangle$. The blue rectangles represent two-qubit parameterized quantum gates of the Quantum Convolutional Neural Network (QCNN), designed for binary classification tasks. (b) Plot depicting the evolution of the trace distance between two ensembles of quantum states embedded by the NQE models during training on the ibmq_toronto device, compared to the trace distance from conventional quantum embedding without NQE. (c) Noiseless QCNN simulation results. (d) The results from QCNN experiments conducted on IBM quantum devices. In (c) and (d), the blue solid, red dashed, and green dash-dotted lines represent the mean training loss histories for conventional ZZ feature embedding, PCA-NQE, and NQE, respectively. The shaded regions in the figure represent one standard deviation from the mean. These values are acquired from five repetitions of each QCNN training with random initialization of parameters. The thicker versions of these lines indicate the theoretical lower bounds for each method.
• Figure 3: Comparative analysis between Neural Quantum Embeddings and Trainable Unitary Embeddings with one, two and three trainable layers. The numerical simulations were conducted under noiseless (left) and noisy (right) environments, utilizing MNIST (top) and Fashion-MNIST (bottom) datasets. For noiseless simulations, we used 1000 iterations, a learning rate of 0.01 learning rate, and batches of 128 data points per iteration. For noisy simulations, we used 200 iterations, a learning rate of 0.05, and batches of 15 data point per iteration. The noisy model simulations utilized the IBM Qiskit FakeGuadalupe environment. The classification accuracies were evaluated using a sample size of 2115 and 2000 data points for the MNIST and Fashion-MNIST datasets, respectively. The mean and one standard deviation from five independent iterations are shown for the loss history.
• Figure 4: The local effective dimension for the circuit with (solid green) and without (dashed blue) NQE. These results are based on ten sets of experiments, each on a distinct artificial dataset with 20 repetitions with random initialization of parameters. The reported values represent the average across all 200 experiments.
• Figure 5: A comparative analysis of the generalization error bound $G$ with varying regularization weights $\lambda$. This plot illustrates the performance enhancement -- lower generalization error bound -- when employing NQE (green triangles) and PCA-NQE (red circles) over conventional methods without NQE (blue squares) in quantum kernel methods. PCA-NQE and NQE were optimized on the ibmq_toronto quantum hardware. The error bound $G$ was determined based on five independent numerical simulations, presenting both the mean and one standard deviation of $G$.