A Resource Efficient Quantum Kernel

Abstract

Quantum processors may enhance machine learning by mapping high-dimensional data onto quantum systems for processing. Conventional feature maps, for encoding data onto a quantum circuit are currently impractical, as the number of entangling gates scales quadratically with the dimension of the dataset and the number of qubits. In this work, we introduce a quantum feature map designed to handle high-dimensional data with a significantly reduced number of qubits and entangling operations. Our approach preserves essential data characteristics while promoting computational efficiency, as evidenced by extensive experiments on benchmark datasets that demonstrate a marked improvement in both accuracy and resource utilization when using our feature map as a kernel for characterization, as compared to state-of-the-art quantum feature maps. Our noisy simulation results, combined with lower resource requirements, highlight our map's ability to function within the constraints of noisy intermediate-scale quantum devices. Through numerical simulations and small-scale implementation on a superconducting circuit quantum computing platform, we demonstrate that our scheme performs on par or better than a set of classical algorithms for classification. While quantum kernels are typically stymied by exponential concentration, our approach is affected with a slower rate with respect to both the number of qubits and features, which allows practical applications to remain within reach. Our findings herald a promising avenue for the practical implementation of quantum machine learning algorithms on near future quantum computing platforms.

Figures (20)

• Figure 1: (a) A CPMap with 8 qubits encoding 15 features. The qubits are then output to be used coherently in any application (including measurement, repetition of the kernel, and input to a quantum neural network). $H$ is the Hadamard gate and $R_z(X_i)$ is a single-qubit rotation around the $z$ axis by angle $X_i$. (b) Diagram of the $C$ unitary. (c) Diagram of the $P$ unitary. (d) An illustration of a quantum CNN with 8 qubits. The two-qubit unitaries $U_1$ and $U_2$ perform convolutions, then the pooling operations are done by measurements $\mathcal{M}$ on one qubit followed by feedforward unitaries $V_1$ and $V_2$ on the other qubit; all such operations involve parameters that must be trained. While (d) uses pooling operations, (a) adopts a similar architectural flow but replaces them with the unitary $P$, allowing more data features to be encoded before the subsequent layer.
• Figure 2: Resource scaling of CPMap vs. ZZFeatureMap (ideal vs. hardware-compiled). The top row reports ideal$/$logical circuit resources (no transpilation) as a function of the number of features: (a) circuit depth, (b) two-qubit gates count (CNOT) operations, and (c) qubit count required by the feature map. The bottom row reports the same quantities after transpiling the circuits to the IBM Torino backend (native gate set and coupling constraints), where the native two-qubit operation is CZ; thus subplot (e) reports CZ counts for the compiled circuits. Across both settings, CPMap exhibits substantially reduced two-qubit resources and depth growth compared to ZZFeatureMap, and the qualitative scaling advantage is preserved after device-aware compilation.
• Figure 3: A circuit for implementing the three-parameter two-qubit gate $N(\alpha, \beta, \gamma)$, requiring three CNOT gates.
• Figure 4: MCC-score-based analysis of different kernels on different datasets scaled to each have 7 features, comparing the effectiveness of each kernel in handling specific types of data. The CPMap (second from right for each dataset) outperforms the ZZFeatureMap (rightmost) on all datasets in this 7-feature benchmark, and is competitive with strong classical kernels (linear, poly, RBF, sigmoid), sometimes exceeding them depending on the dataset. For some datasets, certain kernels completely failed, so there is no bar visible for the ZZFeatureMap for the INS and PD datasets and for the sigmoid kernel for the BCD dataset. Dataset acronyms: Ionosphere (INS), Breast Cancer Diagnostic (BCD), Credit Card Fraud (CC), Parkinson's Disease (PD), Stellar Classification (SC), Heart Disease (HD), and Titanic Survival (T).
• Figure 5: MCC score for the noisy simulation of CPMap on different datasets. Here we used 'fake_ibmq_jakarta' (FIJ), 'fake_ibmq_manila' (FIM) and 'fake_ibmq_perth' (FIP) backends.