Quantum machine learning is an emerging field that combines machine learning with advances in quantum technologies. Many works have suggested great possibilities of using near-term quantum hardware in supervised learning. Motivated by these developments, we present an embedding-based framework for supervised learning with trainable quantum circuits. We introduce both explicit and implicit approaches. The aim of these approaches is to map data from different classes to separated locations in the Hilbert space via a parametrized quantum circuit. We will show that the implicit approach is a generalization of a recently introduced strategy, so-called quantum metric learning. Furthermore, we discuss an intrinsic connection between the explicit approach and those previously proposed quantum classification models. The implicit and explicit approaches, together, provide a unified framework for quantum classification. The utility of our framework is demonstrated by our noise-free and noisy numerical simulations. Moreover, we have conducted classification testing with both implicit and explicit approaches using several IBM Q devices.
Quantum machine learning is an emerging field that combines machine learning with advances in quantum technologies. Many works have suggested great possibilities of using near-term quantum hardware in supervised learning. Motivated by these developments, we present an embedding-based framework for supervised learning with trainable quantum circuits. We introduce both explicit and implicit approaches. The aim of these approaches is to map data from different classes to separated locations in the Hilbert space via a parametrized quantum circuit. We will show that the implicit approach is a generalization of a recently introduced strategy, so-called quantum metric learning. Furthermore, we discuss an intrinsic connection between the explicit approach and those previously proposed quantum classification models. The implicit and explicit approaches, together, provide a unified framework for quantum classification. The utility of our framework is demonstrated by our noise-free and noisy numerical simulations. Moreover, we have conducted classification testing with both implicit and explicit approaches using several IBM Q devices.
Recently, Lloyd et al. [arXiv:2001.03622] introduced \textit{quantum metric learning} as a powerful framework for quantum supervised learning. Such strategy focuses on the training of the quantum embedding, with a goal of `well-separation' of the embedded data in the Hilbert space H. Motivated by such an idea, we present a metric-based framework for supervised learning with trainable quantum circuits. We introduce both explicit and implicit approaches, based on the \textit{metric} defined on the Hilbert space H, to classify data. In particular, with the implicit approach, the number of separated classes (or their labels) in supervised learning problems can be arbitrarily high (with respect to the number of given qubits, or more precisely, the number of measurement), which is an improvement over current quantum supervised learning models, and this approach requires relatively small number of training points. With the explicit approach, we establish an equivalence between our metric-based framework and other quantum supervised learning models. Such equivalence provides a unification of supervised learning frameworks on quantum computer. We also demonstrate our framework by performing both noise-free and noisy numerical simulations and by running the classification on real IBM Q devices.
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.
Machine learning is widely regarded as a promising application in which quantum computing could offer a computational advantage. However, general quantum machine learning models with provable guarantees on performance, trainability, and efficient inference remain limited. In this work, we introduce quantum Hamiltonian embeddings, a supervised learning framework in which a classical training dataset is encoded into a single effective, parameterized Hamiltonian. Training optimizes the parameters of this data-dependent Hamiltonian, whose approximate ground state defines the classification rule. Once the ground state has been approximated using a quantum algorithm, inference can be performed either classically using a classical representation of the learned state, or with a quantum representation of the state. We establish efficient methods for gradient estimation and prove favorable generalization guarantees. Furthermore, under standard cryptographic assumptions, we show that Hamiltonian embeddings can realize hypothesis functions beyond the reach of efficient classical algorithms. We then demonstrate the framework at scale on IBM Heron processors using a standard benchmark dataset. Together, these results establish Hamiltonian embeddings as a new framework for quantum machine learning, combining theoretical guarantees with a path for experimental investigations at scale on quantum hardware.
Quantum classifiers provide sophisticated embeddings of input data in Hilbert space promising quantum advantage. The advantage stems from quantum feature maps encoding the inputs into quantum states with variational quantum circuits. A recent work shows how to map discrete features with fewer quantum bits using Quantum Random Access Coding (QRAC) to encode binary strings into quantum states. We propose a new method to embed discrete features with trainable quantum circuits by combining QRAC and a recently proposed strategy for training quantum feature map called quantum metric learning. The proposed trainable embedding requires not only as few qubits as QRAC but also overcomes the limitations of QRAC to classify inputs whose classes are based on hard Boolean functions. We numerically demonstrate its use in variational quantum classifiers to achieve better performances to classify real-world datasets, and thus its possibility to use near-term quantum computers for machine learning.
Machine Learning (ML) has been widely applied across numerous domains due to its ability to automatically identify informative patterns from data for various tasks. The availability of large-scale data and advanced computational power enables the development of sophisticated models and training strategies, leading to state-of-the-art performance, but it also introduces substantial challenges. Quantum Computing (QC), which exploits quantum mechanisms for computation, has attracted growing attention and significant global investment as it may address these challenges. Consequently, Quantum Machine Learning (QML), the integration of these two fields, has received increasing interest, with a notable rise in related studies in recent years. We are motivated to review these existing contributions regarding quantum circuit-based learning models for classical data analysis and highlight the identified potentials and challenges of this technique. Specifically, we focus not only on QML models, both kernel-based and neural network-based, but also on recent explorations of their integration with classical machine learning layers within hybrid frameworks. Moreover, we examine both theoretical analysis and empirical findings to better understand their capabilities, and we also discuss the efforts on noise-resilient and hardware-efficient QML that could enhance its practicality under current hardware limitations. In addition, we cover several emerging paradigms for advanced quantum circuit design and highlight the adaptability of QML across representative application domains. This study aims to provide an overview of the contributions made to bridge quantum computing and machine learning, offering insights and guidance to support its future development and pave the way for broader adoption in the coming years.
Quantum classifiers provide sophisticated embeddings of input data in Hilbert space promising quantum advantage. The advantage stems from quantum feature maps encoding the inputs into quantum states with variational quantum circuits. A recent work shows how to map discrete features with fewer quantum bits using Quantum Random Access Coding (QRAC), an important primitive to encode binary strings into quantum states. We propose a new method to embed discrete features with trainable quantum circuits by combining QRAC and a recently proposed strategy for training quantum feature map called quantum metric learning. We show that the proposed trainable embedding requires not only as few qubits as QRAC but also overcomes the limitations of QRAC to classify inputs whose classes are based on hard Boolean functions. We numerically demonstrate its use in variational quantum classifiers to achieve better performances in classifying real-world datasets, and thus its possibility to leverage near-term quantum computers for quantum machine learning1.