Supervised learning with quantum enhanced feature spaces (1804.11326v2)

Abstract: Machine learning and quantum computing are two technologies each with the potential for altering how computation is performed to address previously untenable problems. Kernel methods for machine learning are ubiquitous for pattern recognition, with support vector machines (SVMs) being the most well-known method for classification problems. However, there are limitations to the successful solution to such problems when the feature space becomes large, and the kernel functions become computationally expensive to estimate. A core element to computational speed-ups afforded by quantum algorithms is the exploitation of an exponentially large quantum state space through controllable entanglement and interference. Here, we propose and experimentally implement two novel methods on a superconducting processor. Both methods represent the feature space of a classification problem by a quantum state, taking advantage of the large dimensionality of quantum Hilbert space to obtain an enhanced solution. One method, the quantum variational classifier builds on [1,2] and operates through using a variational quantum circuit to classify a training set in direct analogy to conventional SVMs. In the second, a quantum kernel estimator, we estimate the kernel function and optimize the classifier directly. The two methods present a new class of tools for exploring the applications of noisy intermediate scale quantum computers [3] to machine learning.

• The paper introduces a quantum variational classifier that encodes data into quantum states and optimizes circuit parameters for effective supervised learning.
• It presents a quantum kernel estimation method that constructs the kernel matrix directly on a quantum computer for classical SVM classification.
• Experimental results on a superconducting processor demonstrate near-perfect classification, underscoring the practical potential of quantum-enhanced techniques.

Supervised Learning with Quantum Enhanced Feature Spaces

The paper "Supervised learning with quantum enhanced feature spaces" by Vojtech Havlicek et al. proposes a novel framework that combines classical supervised learning techniques with quantum computing principles to address classification problems. The research team demonstrates the feasibility of leveraging the exponential scale of quantum state spaces to enhance machine learning processes, particularly focusing on support vector machines (SVMs).

Summary of Contributions

The paper introduces two primary methodologies:

1. Quantum Variational Classifier: This approach employs a variational quantum circuit to classify data, mimicking the classical SVM approach but leveraging the quantum feature space for enhanced performance.
2. Quantum Kernel Estimation: This method involves estimating the kernel function directly using a quantum computer and subsequently applying classical optimization techniques to determine the classifier.

Quantum Variational Classifier

In the quantum variational classifier approach, the authors build on a concept where data is mapped to a quantum state. A variational quantum circuit $W(\vec{\theta})$ then processes this quantum state, akin to how SVMs map data to a high-dimensional feature space where a hyperplane is constructed to separate labeled samples.

The main steps in this method are as follows:

1. Feature Map Encoding: Classical data $\vec{x} \in \mathbb{R}^d$ is encoded into a quantum state through a unitary transformation ${\cal U}_{\Phi(\vec{x})}$.
2. Quantum Circuit Execution: The variational circuit $W(\vec{\theta})$ is applied to the quantum state, effectively positioning the data within the quantum feature space.
3. Measurement and Classification: Based on the outcomes of measurements performed on the resultant state, a classical label is assigned using a decision rule.
4. Optimization: Parameters of the variational circuit are optimized using classical methods. The empirical risk and probabilities of misclassification guide the optimization routine.

The experiments implement this approach on a superconducting quantum processor, varying the depth of the quantum circuit and observing the classification success. The reported success ratio improves with increased circuit depth, demonstrating the efficacy of the method.

Quantum Kernel Estimation

The quantum kernel estimation method directly utilizes the quantum computer to compute the essential inner products of the feature vectors, forming the kernel matrix required for classical SVM classification.

Key steps include:

1. Kernel Matrix Construction: Estimating $$K(\vec{x}_i,\vec{x}_j) = |{\Phi(\vec{x}_i)}{\Phi(\vec{x}_j)}|^2$$ for all samples in the training set using a quantum circuit.
2. Support Vector Determination: Solving the dual formulation of the SVM optimization problem to find the optimal support vectors and Lagrange multipliers.
3. Classification of New Data: Estimating the kernel between new data points and the support vectors on the quantum computer and using the classical SVM decision function to classify new samples.

The authors emphasize the need for complex feature maps that are challenging to simulate classically to achieve a quantum advantage. In their experiments, they demonstrate nearly perfect classification results, validating the potential of this method.

Implications and Future Directions

The practical implications of this research are substantial, as it paves the way for employing near-term quantum devices for machine learning tasks. The proposed methods are especially relevant for applications requiring high-dimensional feature mapping, where conventional SVMs face computational limitations.

Theoretically, this work underscores the symbiotic relationship between machine learning and quantum computing. It opens avenues for further exploration of quantum algorithms that can outperform classical counterparts on specific tasks.

Future research could focus on refining the feature maps to ensure quantum advantage while maintaining efficiency. Exploring other machine learning models and extending the framework to unsupervised learning and reinforcement learning also present compelling directions for subsequent studies.

Conclusion

This paper represents a significant step forward in the integration of quantum computing with machine learning. By proposing and implementing quantum-enhanced SVM-like classifiers, the authors have contributed valuable methodologies that leverage quantum computational power for supervised learning tasks. The experimental results on a superconducting quantum processor underscore the practical feasibility of these approaches and highlight their potential to revolutionize computational paradigms in machine learning.