Quantum kernel learning (QKL) promises efficient machine learning by encoding feature maps onto exponentially large Hilbert spaces inherent in quantum systems. Using the liquid-state nuclear magnetic resonance (NMR) platform, we implement and benchmark QKL for one-dimensional regression and two-dimensional classification tasks. We then classify entangling and non-entangling operators by extending QKL to handle parametrized or non-parameterized operator inputs. We first compute the kernel numerically for a double-layered star system and then experimentally validate it on a 3-qubit NMR register. QKL provides a practical route to compare operators on native quantum hardware without expensive tomography protocols. Our results confirm the superiority of QKL over other classical methods for processing quantum data, thereby highlighting its ability to capture the inherent structure of quantum space and to extend its domain of operation beyond the training domain by exploiting symmetries in the operator space.
Quantum kernel learning (QKL) promises efficient machine learning by encoding feature maps onto exponentially large Hilbert spaces inherent in quantum systems. Using the liquid-state nuclear magnetic resonance (NMR) platform, we implement and benchmark QKL for one-dimensional regression and two-dimensional classification tasks. We then classify entangling and non-entangling operators by extending QKL to handle parametrized or non-parameterized operator inputs. We first compute the kernel numerically for a double-layered star system and then experimentally validate it on a 3-qubit NMR register. QKL provides a practical route to compare operators on native quantum hardware without expensive tomography protocols. Our results confirm the superiority of QKL over other classical methods for processing quantum data, thereby highlighting its ability to capture the inherent structure of quantum space and to extend its domain of operation beyond the training domain by exploiting symmetries in the operator space.
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.
Quantum-enhanced machine learning is a rapidly evolving field that aims to leverage the unique properties of quantum mechanics to enhance classical machine learning. However, the practical applicability of these methods remains an open question, particularly beyond the context of specifically crafted toy problems, and given the current limitations of quantum hardware. This study focuses on quantum kernel methods in the context of classification tasks. In particular, it examines the performance of quantum kernel estimation and quantum kernel training (QKT) in connection with two quantum feature mappings, namely, ZZFeatureMap and CovariantFeatureMap. Remarkably, these feature maps have been proposed in the literature under the conjecture of possible near-term quantum advantage and have shown promising performance in ad hoc datasets. This study aims to evaluate their versatility and generalization capabilities in a more general benchmark, encompassing both artificial and established reference datasets. Classical machine learning methods, specifically support vector machines and logistic regression, are also incorporated as baseline comparisons. Experimental results indicate that quantum methods exhibit varying performance across different datasets. Despite outperforming classical methods in ad hoc datasets, mixed results are obtained for the general case among standard classical benchmarks. The experimental data call into question a general added value of applying QKT optimization, for which the additional computational cost does not necessarily translate into improved classification performance. Instead, it is suggested that a careful choice of the quantum feature map in connection with proper hyperparameterization may prove more effective.
A basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely, to efficiently perform computations in an intractably large Hilbert space. In this Letter we explore some theoretical foundations of this link and show how it opens up a new avenue for the design of quantum machine learning algorithms. We interpret the process of encoding inputs in a quantum state as a nonlinear feature map that maps data to quantum Hilbert space. A quantum computer can now analyze the input data in this feature space. Based on this link, we discuss two approaches for building a quantum model for classification. In the first approach, the quantum device estimates inner products of quantum states to compute a classically intractable kernel. The kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we use a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. We illustrate these ideas with a feature map based on squeezing in a continuous-variable system, and visualize the working principle with two-dimensional minibenchmark datasets.
We implement an all-optical setup demonstrating kernel-based quantum machine learning for two-dimensional classification problems. In this hybrid approach, kernel evaluations are outsourced to projective measurements on suitably designed quantum states encoding the training data, while the model training is processed on a classical computer. Our two-photon proposal encodes data points in a discrete, eight-dimensional feature Hilbert space. In order to maximize the application range of the deployable kernels, we optimize feature maps towards the resulting kernels' ability to separate points, i.e., their "resolution," under the constraint of finite, fixed Hilbert space dimension. Implementing these kernels, our setup delivers viable decision boundaries for standard nonlinear supervised classification tasks in feature space. We demonstrate such kernel-based quantum machine learning using specialized multiphoton quantum optical circuits. The deployed kernel exhibits exponentially better scaling in the required number of qubits than a direct generalization of kernels described in the literature.
Classical machine learning, extensively utilized across diverse domains, faces limitations in speed, efficiency, parallelism, and processing of complex datasets. In contrast, quantum machine learning algorithms offer significant advantages, including exponentially faster computations, enhanced data handling capabilities, inherent parallelism, and improved optimization for complex problems. In this study, we used the entanglement-enhanced quantum kernel in quantum support vector machine to train complex respiratory data sets. Compared to classical algorithms, our findings reveal that QSVM performs better with 45% higher accuracy for complex respiratory data sets while maintaining comparable performance with linear datasets in contrast to their classical counterparts executed on a 2-qubit system. Through our study, we investigate the efficacy of the QSVM-Kernel algorithm in harnessing the enhanced dimensionality of the quantum Hilbert space for effectively training complex datasets.
Abstract The kernel trick allows us to employ high-dimensional feature space for a machine learning task without explicitly storing features. Recently, the idea of utilizing quantum systems for computing kernel functions using interference has been demonstrated experimentally. However, the dimension of feature spaces in those experiments have been smaller than the number of data, which makes them lose their computational advantage over explicit method. Here we show the first experimental demonstration of a quantum kernel machine that achieves a scheme where the dimension of feature space greatly exceeds the number of data using 1 H nuclear spins in solid. The use of NMR allows us to obtain the kernel values with single-shot experiment. We employ engineered dynamics correlating 25 spins which is equivalent to using a feature space with a dimension over 10 15 . This work presents a quantum machine learning using one of the largest quantum systems to date.