Quantum Machine Learning in Feature Hilbert Spaces.

By Maria Schuld, Nathan Killoran. Physical Review Letters, 2018

1,749 citations recorded by the source corpus.

A basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely, to efficiently perform computations in an intractably large Hilbert space. In this Letter we explore some theoretical foundations of this link and show how it opens up a new avenue for the design of quantum machine learning algorithms. We interpret the process of encoding inputs in a quantum state as a nonlinear feature map that maps data to quantum Hilbert space. A quantum computer can now analyze the input data in this feature space. Based on this link, we discuss two approaches for building a quantum model for classification. In the first approach, the quantum device estimates inner products of quantum states to compute a classically intractable kernel. The kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we use a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. We illustrate these ideas with a feature map based on squeezing in a continuous-variable system, and visualize the working principle with two-dimensional minibenchmark datasets.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/7e7eb0f93c9550d7336f4bbfad5fe89604295705

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

Citations in the corpus

Evaluating analytic gradients on quantum hardware (2018, background): “Examples of such algorithms are variational quantum eigensolvers [1], quantum approximate optimization [2], variational autoencoders [3], quantum feature embeddings [4, 5] and variational classiﬁers [6, 7], but also more general hybrid optimization frameworks [8].”
General parameter-shift rules for quantum gradients (2021, background): “…as optimization [8], quantum chemistry [9, 10, 11, 12, 13], integer factorization [14], compilation [15], quantum control [16], matrix diagonaliza-David Wierichs: wierichs@thp.uni-koeln.de tion [17, 18], and variational quantum machine learning [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].”
Continuous-variable quantum neural networks (2018, methodology): “This scheme can be thought of as an embedding of classical data into a quantum feature space [10].”
Robust data encodings for quantum classifiers (2020, methodology): “[39] M.”
Parameterized quantum circuits as machine learning models (2019, background): “We now briefly discuss both, and refer the Reader to Schuld and Killoran [69] for a more in-depth theoretical exposition.”
Supervised learning with quantum-enhanced feature spaces (2018, methodology): “During composition of this manuscript we became aware of the independent theoretical work by Schuld et al. [29, 30].”
Challenges and opportunities in quantum machine learning (2022, background): “[21] Maria Schuld and Nathan Killoran, “Quantum machine learning in feature hilbert spaces,” Physical review letters 122, 040504 (2019).”
Effect of data encoding on the expressive power of variational quantum-machine-learning models (2020, background): “[24] Maria Schuld and Nathan Killoran, “Quantum machine learning in feature Hilbert spaces,” Physical Review Letters 122, 040504 (2019).”
Variational quantum algorithms (2020, methodology): “In this sense, the unitary V (x) can be used as a quantum non-linear feature map, where the Hilbert space can be exploited for a feature space [134, 135].”
Quantum convolutional neural network for classical data classification (2021, methodology): “In quantum computing, the same analogy can be applied to perform a quantum feature map, which acts as φ : X → H where the vector space H is a Hilbert space [34].”
Strawberry Fields: A Software Platform for Photonic Quantum Computing (2018, neutral): “[34] M.”
Quantum machine learning models are kernel methods (2021, methodology): “The “quantum support vector machine” in [4, 5] is therefore not one of many ideas to build a hybrid classiﬁer, it is a generic blueprint of how to train quantum models in a kernel-based manner.”