Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2001.03622

Quantum embeddings for machine learning

Seth Lloyd, Maria Schuld, Aroosa Ijaz, Josh Izaac, Nathan Killoran

Published Jan 10, 2020• 198 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2001.03622

Related papers

Invited: Trainable Discrete Feature Embeddings for Quantum Machine Learning

Napat Thumwanit, Chayaphol Lortaraprasert, Rudy Raymond

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.

ACM/IEEE Design Automation Conference (DAC)•Nov 8, 2021•11 citations

A Unified Classification Framework with Quantum Metric Learning

Nhat A. Nghiem, Samuel Yen-Chi Chen, Tzu-Chieh Wei

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 $\mathcal{H}$. 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 $\mathcal{H}$, 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.

•Oct 25, 2020•6 citations

Trainable Discrete Feature Embeddings for Quantum Machine Learning

Napat Thumwanit, Chayaphol Lortaraprasert, Hiroshi Yano +1 more

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.

IEEE International Conference on Quantum Computing and Engineering (QCE)•Oct 1, 2021•19 citations

Unified framework for quantum classification

Nhat A. Nghiem, Samuel Yen-Chi Chen, Tzu-Chieh Wei

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.

arXiv:2010.13186•Physical Review Research•Jul 16, 2021•27 citations

Generalization Performance of Quantum Metric Learning Classifiers

Jonathan Kim, Stefan Bekiranov

Quantum computing holds great promise for a number of fields including biology and medicine. A major application in which quantum computers could yield advantage is machine learning, especially kernel-based approaches. A recent method termed quantum metric learning, in which a quantum embedding which maximally separates data into classes is learned, was able to perfectly separate ant and bee image training data. The separation is achieved with an intrinsically quantum objective function and the overall approach was shown to work naturally as a hybrid classical-quantum computation enabling embedding of high dimensional feature data into a small number of qubits. However, the ability of the trained classifier to predict test sample data was never assessed. We assessed the performance of quantum metric learning on test ants and bees image data as well as breast cancer clinical data. We applied the original approach as well as variants in which we performed principal component analysis (PCA) on the feature data to reduce its dimensionality for quantum embedding, thereby limiting the number of model parameters. If the degree of dimensionality reduction was limited and the number of model parameters was constrained to be far less than the number of training samples, we found that quantum metric learning was able to accurately classify test data.

Biomolecules•Oct 27, 2022•3 citations

Trainable Discrete Feature Embeddings for Variational Quantum Classifier

Napat Thumwanit, Chayaphol Lortaraprasert, Hiroshi Yano +1 more

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 learning.

arXiv:2106.09415•Jun 17, 2021•3 citations
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.
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.
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\mathcal{H}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\mathcal{H}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 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.
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 computing holds great promise for a number of fields including biology and medicine. A major application in which quantum computers could yield advantage is machine learning, especially kernel-based approaches. A recent method termed quantum metric learning, in which a quantum embedding which maximally separates data into classes is learned, was able to perfectly separate ant and bee image training data. The separation is achieved with an intrinsically quantum objective function and the overall approach was shown to work naturally as a hybrid classical-quantum computation enabling embedding of high dimensional feature data into a small number of qubits. However, the ability of the trained classifier to predict test sample data was never assessed. We assessed the performance of quantum metric learning on test ants and bees image data as well as breast cancer clinical data. We applied the original approach as well as variants in which we performed principal component analysis (PCA) on the feature data to reduce its dimensionality for quantum embedding, thereby limiting the number of model parameters. If the degree of dimensionality reduction was limited and the number of model parameters was constrained to be far less than the number of training samples, we found that quantum metric learning was able to accurately classify test data.
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 learning.