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arXiv:2102.08991 · PRX Quantum

Generalization in Quantum Machine Learning: A Quantum Information Standpoint

Leonardo Banchi, Jason L. Pereira, Stefano Pirandola

Published Nov 1, 2021• 154 citations
arXivHTMLPDF

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 I 2 (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.

DOI
10.1103/prxquantum.2.040321

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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 I 2 (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.
A key component of a quantum machine learning model operating on classical inputs is the design of an embedding circuit mapping inputs to a quantum state. This paper studies a transfer learning setting in which classical-to-quantum embedding is carried out by an arbitrary parametric quantum circuit that is pre-trained based on data from a source task. At run time, a binary quantum classifier of the embedding is optimized based on data from the target task of interest. The average excess risk, i.e., the optimality gap, of the resulting classifier depends on how (dis)similar the source and target tasks are. We introduce a new measure of (dis)similarity between the binary quantum classification tasks via the trace distances. An upper bound on the optimality gap is derived in terms of the proposed task (dis)similarity measure, two Rényi mutual information terms between classical input and quantum embedding under source and target tasks, as well as a measure of complexity of the combined space of quantum embeddings and classifiers under the source task. The theoretical results are validated on a simple binary classification example.
In a manner analogous to their classical counterparts, quantum classifiers are vulnerable to adversarial attacks that perturb their inputs. A promising countermeasure is to train the quantum classifier by adopting an attack-aware, or adversarial, loss function. This paper studies the generalization properties of quantum classifiers that are adversarially trained against bounded-norm white-box attacks. Specifically, a quantum adversary maximizes the classifier's loss by transforming an input stateρ(x)\rho(x)ρ(x)into a stateτ\tauτthat isϵ\epsilonϵ-close to the original stateρ(x)\rho(x)ρ(x)in p-Schatten distance. Under suitable assumptions on the quantum embeddingρ(x)\rho(x)ρ(x), we derive novel information-theoretic upper bounds on the generalization error of adversarially trained quantum classifiers forp=1p=1p=1andp=∞p=\inftyp=∞. The derived upper bounds consist of two terms: the first is an exponential function of the 2-Renyi mutual information between classical data and quantum embedding, while the second term scales linearly with the adversarial perturbation sizeϵ\epsilonϵ. Both terms are shown to decrease as1/T1/\sqrt{T}1/T​over the training set sizeTTT. An extension is also considered in which the adversary assumed during training has different parameterspppandϵ\epsilonϵas compared to the adversary affecting the test inputs. Finally, we validate our theoretical findings with numerical experiments for a synthetic setting.
Quantum machine learning offers the ability to change the significant task of classifying high-dimensions data by taking advantage of infinite-dimensional, enriched state spaces granting a net benefit of quantum computation. We offer a variational quantum algorithm that accommodates many use cases in quantum computing to classify large-scale and high-dimensions data in real time and in cloud-all computing environments. We created our unique quantum circuits; we optimized the parameterized gates using either gradient methods or gradients-free methods and we investigated quantum kernels that provide the capability to identify embeddings in Hilbert space. Our experimental results using both customized synthetic data and benchmarks have shown that the variational quantum classifier was able to achieve comparability in accuracy to classical deep learning accuracy, while improving scale and robustness to noise. In addition, after we deployed the application in a cloud environment, we assessed latency, throughput and metrics from each cloud vendor, and we discovered that if the latency of the quantum accelerators can be controlled by a pipeline of optimized data and workloads, then they are eligible to satisfy the real-time requirement. Based on a theoretical and practical rapidly emerging basis, these results have indicated that variation quantum classifiers are a pathway for future dataintensive applications deployed in the quantum-cloud ecosystem.
Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum information-theoretic quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.
Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum information-theoretic quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.
In this paper we investigate the connection between quantum information theory and machine learning. In particular, we show how quantum state discrimination can represent a useful tool to address the standard classification problem in machine learning. Previous studies have shown that the optimal quantum measurement theory developed in the context of quantum information theory and quantum communication can inspire a new binary classification algorithm that can achieve higher inference accuracy for various datasets. Here we propose a model for arbitrary multiclass classification inspired by quantum state discrimination, which is enabled by encoding the data in the space of linear operators on a Hilbert space. While our algorithm is quantum-inspired, it can be implemented on classical hardware, thereby permitting immediate applications.