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arXiv:2107.06406 · IEEE International Symposium on Information Theory (ISIT)

A Theoretical Framework for Learning from Quantum Data

Mohsen Heidari, Arun Padakandla, Wojciech Szpankowski A

Published Jul 12, 2021• 9 citations
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Abstract

Over decades traditional information theory of source and channel coding advances toward learning and effective extraction of information from data. We propose to go one step further and offer a theoretical foundation for learning classical patterns from quantum data. However, there are several roadblocks to lay the groundwork for such a generalization. First, classical data must be replaced by a density operator over a Hilbert space. Hence, deviated from problems such as state tomography, our samples are i.i.d density operators. The second challenge is even more profound since we must realize that our only interaction with a quantum state is through a measurement which - due to no-cloning quantum postulate - loses information after measuring it. With this in mind, we present a quantum counterpart of the well-known probably approximately correct (PAC) framework. Based on that, we propose a quantum analogous of the Empirical Risk Minimization (ERM) algorithm for learning measurement hypothesis classes. Then, we establish upper bounds on the quantum sample complexity quantum concept classes.

DOI
10.1109/isit45174.2021.9517721

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Over decades traditional information theory of source and channel coding advances toward learning and effective extraction of information from data. We propose to go one step further and offer a theoretical foundation for learning classical patterns from quantum data. However, there are several roadblocks to lay the groundwork for such a generalization. First, classical data must be replaced by a density operator over a Hilbert space. Hence, deviated from problems such as state tomography, our samples are i.i.d density operators. The second challenge is even more profound since we must realize that our only interaction with a quantum state is through a measurement which - due to no-cloning quantum postulate - loses information after measuring it. With this in mind, we present a quantum counterpart of the well-known probably approximately correct (PAC) framework. Based on that, we propose a quantum analogous of the Empirical Risk Minimization (ERM) algorithm for learning measurement hypothesis classes. Then, we establish upper bounds on the quantum sample complexity quantum concept classes.
This paper studies quantum supervised learning for classical inference from quantum states. In this model, a learner has access to a set of labeled quantum samples as the training set. The objective is to find a quantum measurement that predicts the label of the unseen samples. The hardness of learning is measured via sample complexity under a quantum counterpart of the well-known probably approximately correct (PAC). Quantum sample complexity is expected to be higher than classical one, because of the measurement incompatibility and state collapse. Recent efforts showed that the sample complexity of learning a finite quantum concept classC\mathcal{C}Cscales asO(∣C∣)O(\vert \mathcal{C}\vert)O(∣C∣). This is significantly higher than the classical sample complexity that grows logarithmically with the class size. This work improves the sample complexity bound toO(VC∗log⁡∣C∗∣)O\left(V_{\mathcal{C}^*} \log \left\vert\mathcal{C}^*\right\vert\right)O(VC∗​log∣C∗∣), whereC∗\mathcal{C}^*C∗is the set of extreme points of the convex closure ofC\mathcal{C}CandVC∗V_{\mathcal{C}^*}VC∗​is the shadow-norm of this set. We show the tightness of our bound for the class of bounded Hilbert-Schmidt norm, scaling asO(log⁡∣C∗∣)O\left(\log \left\vert\mathcal{C}^*\right\vert\right)O(log∣C∗∣). Our approach is based on a new quantum empirical risk minimization (ERM) algorithm equipped with a shadow tomography method.
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.
Statistical learning theory provides the mathematical foundations for understanding generalization from finite data. As quantum computing and quantum machine learning continue to develop, there is increasing interest in whether analogous principles can be formulated for learning settings involving quantum data, quantum models, or quantum computation. This paper offers an introductory perspective on quantum statistical learning theory. We begin by recalling the basic ideas of classical statistical learning theory, including empirical risk minimization, hypothesis classes, generalization, and sample complexity, and then discuss how these concepts must be adapted in the quantum setting. In particular, we distinguish between learning with classical data using quantum algorithms, learning with quantum models, and learning directly from quantum data. We then outline the central learning-theoretic objects in the quantum setting, including quantum hypotheses, measurement-based predictions, population and empirical risk, shot complexity, and information-extraction limitations. A simple illustrative example is used to highlight how finite measurements and quantum uncertainty affect learning performance. We conclude with a discussion of open problems and directions toward a more unified theory of generalization and learnability in quantum learning.
We study learning from quantum data, in particular quantum state classification which has applications, among others, in classifying the separability of quantum states. In this learning model, there are nnn quantum states with classical labels as the training samples. Predictors are quantum measurements that when applied to the next unseen quantum state predict its classical label. By integrating learning theory with quantum information, we introduce a quantum counterpart of the PAC framework for learning with respect to classes of measurements. We argue that major challenges arising from the quantum nature of the problem are measurement incompatibility and the no-cloning principle -- prohibiting sample reuse. Then, after introducing a Fourier expansion through Pauli's operators, we study learning with respect to an infinite class of quantum measurements whose operator's Fourier spectrum is concentrated on low degree terms. We propose a quantum learning algorithm and show that the quantum sample complexity depends on the ``compatibility structure of such measurement classes -- the more compatible the class is, the lower the quantum sample complexity will be. We further introduce kkk-junta measurements as a special class of low-depth quantum circuits whose Fourier spectrum is concentrated on low degrees.
Learning with data in the form of quantum states is of current interest and has led to a variety of problems that boil down to interaction with the available data via quantum measurement and classical post-processing of observed classical outcomes. In quantum measurement PAC learning, one is given a sequence of unknown, prepared quantum states and classical labels, along with a hypothesis class of candidate measurements. The task is to select a measurement from the hypothesis class that minimizes a fixed notion of error in prediction of the classical labels via measurement of a new state by the selected hypothesis. In this work, we consider the advantage of interacting with the given data in the measurement learning framework using learning rules given by measurements that cannot be implemented using local operations and classical communication (LOCC), as opposed to single-copy learning rules. We provide a construction showing that there exist learning scenarios wherein single-copy learning rules are asymptotically suboptimal compared to optimal ones. We then show that learning rules based on entangled measurements enjoy at most a polynomial sample complexity advantage over single-copy learning rules in the PAC learning setting (under a natural joint measurability covering assumption).