Quantum embeddings for machine learning (2001.03622v2)

Abstract: Quantum classifiers are trainable quantum circuits used as machine learning models. The first part of the circuit implements a quantum feature map that encodes classical inputs into quantum states, embedding the data in a high-dimensional Hilbert space; the second part of the circuit executes a quantum measurement interpreted as the output of the model. Usually, the measurement is trained to distinguish quantum-embedded data. We propose to instead train the first part of the circuit -- the embedding -- with the objective of maximally separating data classes in Hilbert space, a strategy we call quantum metric learning. As a result, the measurement minimizing a linear classification loss is already known and depends on the metric used: for embeddings separating data using the l1 or trace distance, this is the Helstrom measurement, while for the l2 or Hilbert-Schmidt distance, it is a simple overlap measurement. This approach provides a powerful analytic framework for quantum machine learning and eliminates a major component in current models, freeing up more precious resources to best leverage the capabilities of near-term quantum information processors.

• The paper presents quantum metric learning by optimizing quantum feature maps to maximize class separation in high-dimensional Hilbert spaces.
• It demonstrates that simpler quantum circuits, such as the SWAP test, efficiently measure data overlap, enhancing classification accuracy.
• The study reduces circuit complexity by eliminating the need to train measurement bases, paving the way for more resource-efficient hybrid quantum classifiers.

Quantum Embeddings for Machine Learning: A Technical Overview

The paper "Quantum embeddings for machine learning" by Seth Lloyd and collaborators explores the field of quantum machine learning, presenting a novel approach known as quantum metric learning. The authors propose a distinctive strategy to improve the efficiency and effectiveness of quantum classifiers, which are a specific type of machine learning model executed in quantum computing environments. These classifiers leverage the high-dimensionality of quantum systems combined with the probabilistic nature of quantum mechanics to solve classification tasks.

Overview of Quantum Classifiers

Traditional quantum classifiers operate by first encoding classical input data into quantum states using a quantum feature map, effectively embedding the data into a high-dimensional Hilbert space. The classification is performed by quantum measurements that distinguish between classes in the encoded space. Normally, variations in the classifier involve training the measurement basis to optimize class separation. However, the paper challenges this paradigm by suggesting that it is more effective to optimize the quantum feature map itself—the embedding—thus maximizing class separation within the Hilbert space directly.

Quantum Metric Learning

The approach termed quantum metric learning focuses on adapting the initial embedding to maximize the distance between classes in the Hilbert space. This is a departure from previous methods where the measurement aspects of the circuit were the main focus of training. In this context, the optimal measurement is inherently known: the Helstrøm measurement is optimal for data separated by trace distance, while a simple overlap measurement suffices when the data is separated by Hilbert-Schmidt distance.

Implementation and Theoretical Implications

The paper presents practical strategies for implementing these measurements on near-term quantum devices. When training an embedding to optimize for the $\ell_2$ (Hilbert-Schmidt) distance, the authors spotlight the use of simple quantum circuits like the SWAP test, which can efficiently measure data overlaps. Both $\ell_1$ (trace) and $\ell_2$ distances are quantitatively analyzed, demonstrating that their optimized embeddings result in efficient classifiers without needing deep quantum circuits.

Impact and Future Developments

From a practical perspective, this approach eliminates the need for complex quantum circuits to train the measurement section of the classifier. This reduction in complexity is crucial for the nascent field of quantum computing, making more effective use of limited quantum resources. Theoretically, this research opens up new avenues for exploring quantum feature maps as a kernel-like method in quantum machine learning, where the feature map's power becomes the central element of differentiation against classical methods.

The implications of this paper extend to the training efficiency and representational capacity of quantum models, potentially offering significant advantages for datasets where classical separability may be weak or costly to achieve. Looking ahead, quantum metric learning may well stimulate advancements in hybrid quantum-classical architectures, where quantum embeddings are utilized within classically computational frameworks, akin to how kernel methods are used alongside support vector machines.

In summary, "Quantum embeddings for machine learning" sets a foundation for more resource-efficient quantum machine learning models by leveraging the unique properties of quantum systems to achieve separability through metric learning, thus paving the way for advanced applications as quantum computing technology evolves.