Neural auto-designer for enhanced quantum kernels

TL;DR

This work addresses the challenge of designing effective quantum feature maps for real-world data on noisy intermediate-scale quantum (NISQ) devices. It reframes kernel design as a discrete-continuous optimization and introduces QuKerNet, which uses Max-Relevance Min-Redundancy feature selection to manage high-dimensional data and a neural predictor trained on kernel-target alignment (KTA) to efficiently rank candidate circuit layouts before fine-tuning encoder parameters. The approach jointly optimizes gate layout $S$ and parameters $\bm{\theta}$, enabling automatic discovery of problem-specific quantum kernels that outperform classical baselines and prior quantum kernels on multiple datasets, while mitigating kernel concentration and showing robustness to noise. The results demonstrate that integrating deep learning with quantum kernel design can unlock practical advantages for real-world tasks on NISQ hardware, underscoring a significant step forward in data-driven quantum machine learning.

Abstract

Quantum kernels hold great promise for offering computational advantages over classical learners, with the effectiveness of these kernels closely tied to the design of the quantum feature map. However, the challenge of designing effective quantum feature maps for real-world datasets, particularly in the absence of sufficient prior information, remains a significant obstacle. In this study, we present a data-driven approach that automates the design of problem-specific quantum feature maps. Our approach leverages feature-selection techniques to handle high-dimensional data on near-term quantum machines with limited qubits, and incorporates a deep neural predictor to efficiently evaluate the performance of various candidate quantum kernels. Through extensive numerical simulations on different datasets, we demonstrate the superiority of our proposal over prior methods, especially for the capability of eliminating the kernel concentration issue and identifying the feature map with prediction advantages. Our work not only unlocks the potential of quantum kernels for enhancing real-world tasks but also highlights the substantial role of deep learning in advancing quantum machine learning.

Figures (14)

• Figure 1: The workflow of QuKerNet. In step 1, QuKerNet sets up the search space $\mathcal{S}$ via the accessible basic quantum gate set $\mathcal{G}$. For example, the set of single-qubit gates includes $R_X$, $R_Y$, $R_Z$ represented by colored hexagons, and the set of two-qubit gates contains $\rm{CNOT, CZ, SWAP}$ represented by the colored rectangles. In step 2, a pentagram represents a feasible candidate circuit in the search space $\mathcal{S}$. In the training process, $M$ candidate circuits (highlighted by the red pentagrams) are collected and transformed to an image via the proposed encoding method. Meanwhile, KTA of these sampled candidate circuits is calculated according to Eq. (\ref{['KTA']}). These training data are used to train an MLP-based neural predictor. In step 3, the optimized neural predictor is employed to predict the performance of a large number of candidate circuits sampled from the search space $\mathcal{S}$. Afterward, QuKerNet ranks the predicted KTA and selects the top $k$ candidate circuits. In step 4, for each candidate circuit, QuKerNet randomly replaces $m$ parameterized gates, which are used to encode data features, by the variational parameters $\bm{\theta}$, highlighted by the hexagon with shadow. These parameters are optimized to maximize KTA. Then we compare their classification accuracy and the circuit with the highest accuracy is selected as the output circuit.
• Figure 2: Role of feature selection of QuKerNet. (a) The depiction of hardware efficient ansatz (HEA). Each layer is composed of $R_X$ gates where the rotational angle on the $i$-th qubit amounts to the $i$-th feature of the input data, and CNOT gates acting on the adjacent qubits. (b) Alleviation of vanishing similarity by feature selection. The label of "KV" represents kernel variance. Both "F" and 'dimensions' refer to the feature dimension. (c) Comparison with two feature selection methods, i.e., mRMR and random selection.
• Figure 3: Correlation of kernels optimized under different loss functions.
• Figure 4: Numerical results of QuKerNet. (a) The best kernels discovered by TEK, QuKerNet-1, and QuKerNet-2 on tailored MNIST and tailored CC datasets. (b) The training and test accuracy for the best kernels searched by TEK, QuKerNet-1, and QuKerNet-2 on the synthetic dataset. Note that the results of HEAK have no standard deviation as it is a fixed kernel without any adjustable parameters.
• Figure 5: HEAK layout and TEK layout. The $\bm{x}_j$ is corresponding to the $j\text{-th}$ feature in sample $\hat{\bm{x}}^{(i)}$. (a) The layout of HEAK. And each dashed region is a block that is used to encode up to $N$ features of $\hat{\bm{x}}^{(i)}$. (b) The layout of TEK. The layout of the dashed region is used for encoding data, while the solid line portion represents variational parameters used to adjust the performance of the kernel. A trainable module is connected after each block of HEAK. The dashed area and the solid line area together form a block.