Power of data in quantum machine learning

Huang, Hsin-Yuan; Broughton, Michael; Mohseni, Masoud; Babbush, Ryan; Boixo, Sergio; Neven, Hartmut; McClean, Jarrod R.

doi:10.1038/s41467-021-22539-9

Cited by 389 publications

(393 citation statements)

References 58 publications

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“…Given the fundamental role of generalization bounds, there has recently been a strong and steady stream of works contributing to the derivation of generalization bounds for PQC-based models [24][25][26][27][28][29][30][31][32]. However, as discussed in detail in Section 4, these prior works all differ from our results in a variety of ways.…”

Section: Introductionmentioning

confidence: 82%

“…Finally, we mention Ref. [30] which has developed techniques for evaluating the potential advantages of quantum kernels over classical kernels. These results are of relevance to this work due to the close relationship between PQC-based models and kernel methods [16].…”

Section: Encoding-dependent Complexity and Generalization Boundsmentioning

confidence: 99%

“…In a first step, the authors of Ref. [30] suggest the evaluation of a geometric quantity which depends on the chosen quantum feature map and the available training data instances. If the quantum machine learning model passes this first test, a model complexity parameter, which now depends on the quantum encoding and the training data (both instances and labels), should be computed.…”

Section: Encoding-dependent Complexity and Generalization Boundsmentioning

confidence: 99%

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Encoding-dependent generalization bounds for parametrized quantum circuits

Matthias

Gil-Fuster

Meyer

et al. 2021

Quantum

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A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.

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Section: Introductionmentioning

confidence: 82%

Section: Encoding-dependent Complexity and Generalization Boundsmentioning

confidence: 99%

Section: Encoding-dependent Complexity and Generalization Boundsmentioning

confidence: 99%

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Encoding-dependent generalization bounds for parametrized quantum circuits

Matthias

Gil-Fuster

Meyer

et al. 2021

Quantum

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“…The concept of a coreset opens a new paradigm for training ML models by using small quantum computers [9,10] since currently available quantum computers offered by D-Wave Systems (D-Wave QA) and by IBM quantum experience (a gate-based quantum computer) comprise very few quantum bits (qubits) (https://cloud.dwavesys.com/leap, https://quantum-computing.ibm.com/, accessed on 30 August 2021). In particular, quantum computers promise to solve some intractable problems in ML [11][12][13], and to train an SVM even better/faster than a conventional computer when its input data volume is very small ("core of a dataset") [14,15]. Training ML methods by using a quantum computer or by exploiting quantum information is called Quantum Machine Learning (QML) [16][17][18], and finding the solutions of the SVM on a quantum computer is termed a quantum SVM (qSVM), otherwise classical SVM (cSVM).…”

Section: Introductionmentioning

confidence: 99%

Assembly of a Coreset of Earth Observation Images on a Small Quantum Computer

Otgonbaatar

Datcu

2021

Electronics

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Satellite instruments monitor the Earth’s surface day and night, and, as a result, the size of Earth observation (EO) data is dramatically increasing. Machine Learning (ML) techniques are employed routinely to analyze and process these big EO data, and one well-known ML technique is a Support Vector Machine (SVM). An SVM poses a quadratic programming problem, and quantum computers including quantum annealers (QA) as well as gate-based quantum computers promise to solve an SVM more efficiently than a conventional computer; training the SVM by employing a quantum computer/conventional computer represents a quantum SVM (qSVM)/classical SVM (cSVM) application. However, quantum computers cannot tackle many practical EO problems by using a qSVM due to their very low number of input qubits. Hence, we assembled a coreset (“core of a dataset") of given EO data for training a weighted SVM on a small quantum computer, a D-Wave quantum annealer with around 5000 input quantum bits. The coreset is a small, representative weighted subset of an original dataset, and its performance can be analyzed by using the proposed weighted SVM on a small quantum computer in contrast to the original dataset. As practical data, we use synthetic data, Iris data, a Hyperspectral Image (HSI) of Indian Pine, and a Polarimetric Synthetic Aperture Radar (PolSAR) image of San Francisco. We measured the closeness between an original dataset and its coreset by employing a Kullback–Leibler (KL) divergence test, and, in addition, we trained a weighted SVM on our coreset data by using both a D-Wave quantum annealer (D-Wave QA) and a conventional computer. Our findings show that the coreset approximates the original dataset with very small KL divergence (smaller is better), and the weighted qSVM even outperforms the weighted cSVM on the coresets for a few instances of our experiments. As a side result (or a by-product result), we also present our KL divergence findings for demonstrating the closeness between our original data (i.e., our synthetic data, Iris data, hyperspectral image, and PolSAR image) and the assembled coreset.

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“…However, the gate-based quantum computer itself is posing several new challenges, for instance, how to map classical data to qubits (quantum data) depending on the limited number of its input qubits, or how to use the specificity of the "qubits" to obtain quantum advantages over nonquantum computing techniques, while ubiquitous data in practical domains are of classical nature. In particular, the input data play an important role in a quantum algorithm to obtain quantum advantages, and for example, in scientific studies [9], [10], their authors implied that QML networks achieve quantum advantages over a conventional technique only if classical data are naturally embedded in their input qubits, or their input data are quantum data.…”

mentioning

confidence: 99%

Natural Embedding of the Stokes Parameters of Polarimetric Synthetic Aperture Radar Images in a Gate-Based Quantum Computer

Otgonbaatar

Datcu

2022

IEEE Trans. Geosci. Remote Sensing

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Quantum algorithms are designed to process quantum data (quantum bits) in a gate-based quantum computer. They are proven rigorously that they reveal quantum advantages over conventional algorithms when their inputs are certain quantum data or some classical data mapped to quantum data. However, in a practical domain, data are classical in nature, and they are very big in dimension, size, and so on. Hence, there is a challenge to map (embed) classical data to quantum data, and even no quantum advantages of quantum algorithms are demonstrated over conventional ones when one processes the mapped classical data in a gate-based quantum computer. For the practical domain of earth observation (EO), due to the different sensors on remotesensing platforms, we can map directly some types of EO data to quantum data. In particular, we have polarimetric synthetic aperture radar (PolSAR) images characterized by polarized beams. A polarized state of the polarized beam and a quantum bit are the Doppelganger of a physical state. We map them to each other, and we name this direct mapping a natural embedding, otherwise an artificial embedding. Furthermore, we process our naturally embedded data in a gate-based quantum computer by using a quantum algorithm regardless of its quantum advantages over conventional techniques; namely, we use the QML network as a quantum algorithm to prove that we naturally embedded our data in input qubits of a gate-based quantum computer. Therefore, we employed and directly processed PolSAR images in a QML network. Furthermore, we designed and provided a QML network with an additional layer of a neural network, namely, a hybrid quantum-classical network, and demonstrate how to program (via optimization and backpropagation) this hybrid quantum-classical network when employing and processing PolSAR images. In this work, we used a gate-based quantum computer offered by an IBM Quantum and a classical simulator for a gate-based quantum computer. Our contribution is that we provided very specific EO data with a natural embedding feature, the Doppelganger of quantum bits, and processed them in a hybrid quantum-classical network. More importantly, in the future, these PolSAR data can be processed by future quantum algorithms and future quantum computing platforms to obtain (or demonstrate) some quantum advantages over conventional techniques for EO problems.

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Power of data in quantum machine learning

Cited by 389 publications

References 58 publications

Encoding-dependent generalization bounds for parametrized quantum circuits

Encoding-dependent generalization bounds for parametrized quantum circuits

Assembly of a Coreset of Earth Observation Images on a Small Quantum Computer

Natural Embedding of the Stokes Parameters of Polarimetric Synthetic Aperture Radar Images in a Gate-Based Quantum Computer

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