Abstract
Quantum classification and hypothesis testing (state and channel discrimination) are two tightly related subjects, the main difference being that the former is data driven: how to assign to quantum states ρ(x) the corresponding class c (or hypothesis) is learnt from examples during training, where x can be either tunable experimental parameters or classical data “embedded” into quantum states. Does the model generalize? This is the main question in any data-driven strategy, namely the ability to predict the correct class even of previously unseen states. Here we establish a link between quantum classification and quantum information theory, by showing that the accuracy and generalization capability of quantum classifiers depend on the (Rényi) mutual information I(C:Q) and I2(X:Q) between the quantum state space Q and the classical parameter space X or class space C. Based on the above characterization, we then show how different properties of Q affect classification accuracy and generalization, such as the dimension of the Hilbert space, the amount of noise, and the amount of neglected information from X via, e.g., pooling layers. Moreover, we introduce a quantum version of the information bottleneck principle that allows us to explore the various trade-offs between accuracy and generalization. Finally, in order to check our theoretical predictions, we study the classification of the quantum phases of an Ising spin chain, and we propose the variational quantum information bottleneck method to optimize quantum embeddings of classical data to favor generalization.1 MoreReceived 4 March 2021Accepted 30 September 2021DOI:https://doi.org/10.1103/PRXQuantum.2.040321Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasMachine learningQuantum channelsQuantum computationQuantum cryptographyQuantum feedbackQuantum information processingQuantum opticsQuantum sensingQuantum InformationAtomic, Molecular & OpticalStatistical PhysicsInterdisciplinary Physics
Highlights
Quantum information and machine learning are two very active areas of research that have become increasingly interconnected [1,2,3,4]
III we introduce the main technical result of this paper: quantities linked to either the training or testing errors can be bounded by the quantum mutual information between some suitable quantum states and classical variables
IV we show different implications of our theoretical bounds: we introduce a quantum version of the bias-variance trade-off, which defines fundamental limitations on the testing error for finite amounts of data; we show how to use results developed in the quantum communication and cryptography literature to study how to optimally embed classical information onto quantum states; we show how different properties of the quantum states affect the classification accuracy and generalization, such as the dimension of the Hilbert space, the amount of noise, and the amount of neglected information via, e.g., pooling layers
Summary
Quantum information and machine learning are two very active areas of research that have become increasingly interconnected [1,2,3,4]. The mathematical derivations of our results, as well as the extension to multiary classification, are presented in the appendices
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