Experimental Machine Learning with Classical and Quantum Data via NMR Quantum Kernels

TL;DR

The paper addresses the challenge of learning nonlinear patterns by leveraging quantum kernels that map data into high-dimensional quantum operator spaces. It demonstrates an experimental implementation on a 10-qubit NMR star-topology register, encoding inputs with data-dependent unitaries and measuring kernel values to perform classical tasks such as one-dimensional regression and two-dimensional classification. A double-layered star extension enables handling non-parametrized operator inputs for quantum tasks, including entangling vs non-entangling unitary classification with high accuracy, both numerically and experimentally. Overall, the work provides evidence that quantum kernels can generalize well to unseen data and potentially enable efficient processing of quantum data on NMR platforms, marking a step toward practical quantum machine learning.

Abstract

Kernel methods map data into high-dimensional spaces, enabling linear algorithms to learn nonlinear functions without explicitly storing the feature vectors. Quantum kernel methods promise efficient learning by encoding feature maps into exponentially large Hilbert spaces inherent in quantum systems. In this work, we implement quantum kernels on a 10-qubit star-topology register in a nuclear magnetic resonance (NMR) platform. We experimentally encode classical data in the evolution of multiple quantum coherence orders using data-dependent unitary transformations and then demonstrate one-dimensional regression and two-dimensional classification tasks. By extending the register to a double-layered star configuration, we propose an extended quantum kernel to handle non-parametrized operator inputs. Specifically, we set up a kernel for the classification of entangling and non-entangling operations and then validate this kernel first numerically by computing it on a double-layered star register and then experimentally by computing it on a three-qubit NMR register. Our results show that this kernel exhibits an ability to generalize well over unseen data. These results confirm that quantum kernels possess strong capabilities in classical as well as quantum machine learning tasks.

Figures (6)

• Figure 1: Illustrating data (left) and its feature space mapping using a higher-dimensional kernel (right).
• Figure 2: Quantum feature mapping and quantum kernel computation protocol. The quantum circuit starts with the initial state of $\rho_{eq}^C$ in order to ensure the symmetry of the quantum kernel since only the $C$ qubit is measured at the end.
• Figure 3: (a) Geometry of the star-topology register. (b) Molecular structure of trimethyl phosphite, where $^{31}$P and $^1$H spins form central (C) and (A) ancillary qubits. Here, the spin-spin coupling constant $J_\mathrm{PH} = 11.0$ Hz. (c) Quantum circuit for encoding classical inputs $x_i$ and $x_j$ in terms of unitaries $U(x_i)$ and $U(x_j)$, followed by measurement of $z$ magnetization of C qubit. Here, helix represents the dephasing operations on $x$ and $y$ magnetization of C-qubit. (d) Experimentally obtained kernel function for one-dimensional inputs using the quantum kernel in Eq. \ref{['eq:knmr1d']}. Assuming the symmetry of the kernel function, the left half is obtained by mirroring the experimentally extracted right half.
• Figure 4: Regression of (a) sine function with 15 training data points and (b) a seventh-degree polynomial function with 42 training data points. (c) and (d) show the percentage RMSE as a function of the number of training points used for the sine function and the polynomial function, respectively.
• Figure 5: (a) The $x_j^{(1)}$ slices of the experimentally obtained kernel for two-dimensional input. (b,c) Two-dimensional classifications for (b) circles dataset and (c) moons dataset. In both classifications, the squares and circles represent the training data points belonging to two different classes. The background represents the decision function obtained after training.