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Nature Communications

Understanding quantum machine learning also requires rethinking generalization

Elies Gil-Fuster, Jens Eisert, Carlos Bravo-Prieto

Published Mar 13, 2024• 82 citations
Publisher

Abstract

Quantum machine learning models have shown successful generalization performance even when trained with few data. In this work, through systematic randomization experiments, we show that traditional approaches to understanding generalization fail to explain the behavior of such quantum models. Our experiments reveal that state-of-the-art quantum neural networks accurately fit random states and random labeling of training data. This ability to memorize random data defies current notions of small generalization error, problematizing approaches that build on complexity measures such as the VC dimension, the Rademacher complexity, and all their uniform relatives. We complement our empirical results with a theoretical construction showing that quantum neural networks can fit arbitrary labels to quantum states, hinting at their memorization ability. Our results do not preclude the possibility of good generalization with few training data but rather rule out any possible guarantees based only on the properties of the model family. These findings expose a fundamental challenge in the conventional understanding of generalization in quantum machine learning and highlight the need for a paradigm shift in the study of quantum models for machine learning tasks.

DOI
10.1038/s41467-024-45882-z

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Quantum machine learning is at the crossroads of two of the most exciting current areas of research: quantum computing and classical machine learning. Although the field is still in its infancy, the body of literature is already large enough to warrant several review articles [1–3]. This short survey focuses on a selection of significant recent results on the subtopic of quantum neural networks, an area that hopes to build on the enormous impact that classical neural networks have had. In particular, we concentrate on quantum generalizations of popular neural-network concepts such as Boltzmann machines, generative adversarial networks and autoencoders, as well as applications of classical neural networks to quantum information and vice versa. Examples of Boltzmann machines. In a fully connected Boltzmann machine, connections exist between all visible and hidden neurons. In a restricted Boltzmann machine (RBM), there are no connections between neurons of the same type. A deep Boltzmann machine (DBM) consists of adding one or more additional layers of hidden neurons to an RBM, with no connections between neurons in the same layer. The energy-based nature of BMs gives a natural framework for considering quantum generalizations of their behavior. The BM energy function is equivalent to the Hamiltonian of a simple Ising model and one might hope that more general Hamiltonians allowed by quantum mechanics could explain certain data sets better than classically. Amin et al. [4] adopt this approach, and take the so-called transverse field Ising Hamiltonian as the energy function. This increased generality brings certain difficulties, however, as the transverse field term makes training the quantum Boltzmann machine (QBM) non-trivial. Although the general scaling and performance of QBMs is not yet well understood, for certain small-size examples the authors observe that their QBMs can learn data distributions better than classical BMs. Kieferova and Wiebe [5] extend the formalism of Amin et al. and consider more general Hamiltonians that, unlike the transverse Ising model, are not in general believed to be classically efficiently simulatable. Furthermore, they propose new methods for training that allow for parameters corresponding to quantum terms to be learned. Finally, their QBM enables a form of quantum-state tomography that can learn non-classical data, and also generate copies of the reconstructed state. Again, small-scale numerical evidence suggests that their proposals may provide an advantage over classical BMs, although their tendency to overfit data must be investigated further. The practical value of BMs, either classical or quantum, relies in part on the ease with which they are trained. This in turn depends on sampling efficiently from the Boltzmann distribution. Classically this can be done approximately only for the special case of restricted Boltzmann machines (RBM), i.e. where connections are between visible and hidden units, and no connections exist between neurons of the same type. Wiebe et al. [6] propose two quantum algorithms for Boltzmann sampling, based on starting from a state corresponding to the product distribution that is closest to the target Boltzmann distribution. While small-scale numerical studies are encouraging, true bounds on the running time of their algorithms rely on a problem-dependent parameter κ, the scaling of which is hard to estimate. The hope remains though that empirically their algorithms work well in practice, even for multi-layer and fully connected BMs. The expressive power of BMs for compactly representing complicated probability distributions has led to another line of investigation, namely their use for representing quantum many-body wave functions. The authors apply this technique to two prototypical spin models and achieve high accuracy in both ground-state estimation and dynamical evolution, improving on existing state-of-the-art techniques. Their results indicate that RBM states can capture complex many-body correlations in an efficient manner—evidence reinforced by the work of Deng et al. [8], who show that certain topological quantum states in one, two and three dimensions can be represented by RBM states with a number of parameters only linear in the system size. In a subsequent study, however, Gao and Duan [9] prove that although RBM states can indeed represent many highly entangled states, their expressive power is limited in that they cannot represent arbitrary states generated by polynomial-sized quantum circuits, nor those that are the ground states of k-local Hamiltonians with polynomial-sized energy gaps, unless the polynomial hierarchy from computational complexity theory collapses. On the other hand, they show that these two important classes of states can be efficiently represented by a deep Boltzmann machine of three layers. Intriguingly though, another paper by Deng et al. [10] gives an analytic construction of a family of RBM states with a number of parameters linear in the system size that exhibits maximal volume-law entanglement, and that cannot be represented efficiently by either matrix product states (MPS) or tensor-network (TN) states. This shows that the expressive power of even efficiently parametrized RBM states is not limited by entanglement in the same way that other popular compact representations are. This gives the hope that RBM states may prove a valuable tool in understanding entangled quantum systems intractable by traditional methods. Levine et al. [11] establish another connection between the many-body wave function and neural networks, this time in the form of convolutional arithmetic circuits (ConvAC), a variant of the popular convolutional neural networks in widespread use today. By proving a one-to-one equivalence between a ConvAC circuit and TN quantum states, the authors are able to use tools from quantum information theory to analyze the entanglement present in the corresponding TN, and quantify the neural network's ability to model complex correlations between its inputs. This in turn can be used to obtain bounds on the expressivity of a given ConvAC architecture and be of practical use in choosing network parameters tailored to the data. While ConvAC are not widely used in practice, the authors empirically validate that the insights gained can be applied successfully to more common convolutional neural-network designs. Generative adversarial networks (GAN) are machine-learning algorithms in which two players—a generator and a discriminator—compete against each another. The generator's goal is to create and send fake data samples to the discriminator and maximize the probability that the discriminator erroneously believes that the sample comes from some true data distribution. The discriminator attempts to discern real from fake data as best as possible. Under reasonable assumptions the game converges to a point where the discriminator is unable to discriminate with probability greater than 1/2 (i.e. no better than random guessing), and the generator in fact has learned to generate data from the true distribution. Lloyd and Weedbrook [12] consider the natural extension in which one or both of the generator and discriminator are equipped with quantum information processors. When the data to be learned are a quantum state and both parties have quantum processors, the authors show that the unique end point of the adversarial game also coincides with the generator producing a quantum state corresponding to the true data statistics, and the discriminator unable to discern true from false data better than random guessing. They further argue that a different outcome may hold when the true samples are produced by the measurement of some quantum state, and the generator is purely classical. The concept of quantum supremacy is based on the assumption that there exist probability distributions accessible to a quantum computer that no classical computer can efficiently sample from. If the true data correspond to one of these distributions, then no classical generator can efficiently generate fake data reproducing the true statistics and, in principle, the discriminator can successfully discriminate true from fake data. However, whether the discriminator can efficiently find the measurements required to do so is unclear. Quantum-computing techniques have also found applications in autoencoders—neural networks where a larger number of input and output neurons are connected via a smaller number of intermediary latent neurons. The goal is to tune the weights of the network so that a training set of classical bit strings can be approximately recovered after being passed through the autoencoder. Romero et al. [13] consider instead a training set of quantum states, and ask whether a quantum circuit with a similar bottleneck structure can compress and approximately recover the data. The circuits that they consider consist of gates chosen from some set of parametrized unitary operations, and arranged in some fixed order. The parameters are then optimized classically so that the circuit best predicts the training-set labels. The authors demonstrate the validity of this approach by showing that ground states of the Hubbard model and certain molecules can indeed be compressed and recovered by such circuits. However, in general, the performance of such quantum autoencoders will be highly dependent on the kind of programmable circuit on which they are based, and at present there does not seem to be a structured way of determining this. Quantum neural networks are still in their infancy and, while the above works are important steps towards bridging our understanding between classical neural networks and quantum computing, many questions remain unanswered. If the expressivity of Boltzmann machine states is not restricted by entanglement in the same way that MPS and tensor-network states are, what is the best way of understanding their limitations? How can physicists use the correspondence between ConvAC and TN to their advantage in finding efficient, expressive representations of many-body wave functions? Can classical data distributions be learned more efficiently by quantum GAN than classical GAN? Whatever the outcome of these and many other related questions, this promises to be an exciting and active area of research in the years to come.

National Science Review•Nov 29, 2018•42 citations
Quantum machine learning models have shown successful generalization performance even when trained with few data. In this work, through systematic randomization experiments, we show that traditional approaches to understanding generalization fail to explain the behavior of such quantum models. Our experiments reveal that state-of-the-art quantum neural networks accurately fit random states and random labeling of training data. This ability to memorize random data defies current notions of small generalization error, problematizing approaches that build on complexity measures such as the VC dimension, the Rademacher complexity, and all their uniform relatives. We complement our empirical results with a theoretical construction showing that quantum neural networks can fit arbitrary labels to quantum states, hinting at their memorization ability. Our results do not preclude the possibility of good generalization with few training data but rather rule out any possible guarantees based only on the properties of the model family. These findings expose a fundamental challenge in the conventional understanding of generalization in quantum machine learning and highlight the need for a paradigm shift in the study of quantum models for machine learning tasks.
Quantum machine learning models have shown successful generalization performance even when trained with few data. In this work, through systematic randomization experiments, we show that traditional approaches to understanding generalization fail to explain the behavior of such quantum models. Our experiments reveal that state-of-the-art quantum neural networks accurately fit random states and random labeling of training data. This ability to memorize random data defies current notions of small generalization error, problematizing approaches that build on complexity measures such as the VC dimension, the Rademacher complexity, and all their uniform relatives. We complement our empirical results with a theoretical construction showing that quantum neural networks can fit arbitrary labels to quantum states, hinting at their memorization ability. Our results do not preclude the possibility of good generalization with few training data but rather rule out any possible guarantees based only on the properties of the model family. These findings expose a fundamental challenge in the conventional understanding of generalization in quantum machine learning and highlight the need for a paradigm shift in the study of quantum models for machine learning tasks.
The core of quantum machine learning is to devise quantum models with good trainability and low generalization error bounds than their classical counterparts to ensure better reliability and interpretability. Recent studies confirmed that quantum neural networks (QNNs) have the ability to achieve this goal on specific datasets. In this regard, it is of great importance to understand whether these advantages are still preserved on real-world tasks. Through systematic numerical experiments, we empirically observe that current QNNs fail to provide any benefit over classical learning models. Concretely, our results deliver two key messages. First, QNNs suffer from the severely limited effective model capacity, which incurs poor generalization on real-world datasets. Second, the trainability of QNNs is insensitive to regularization techniques, which sharply contrasts with the classical scenario. These empirical results force us to rethink the role of current QNNs and to design novel protocols for solving real-world problems with quantum advantages.
Quantum machine learning is a promising paradigm for learning from limited data, a central bottleneck in domains such as medical imaging, clinical trials, and rare diseases. Quantum convolutional neural networks (QCNNs) are particularly attractive in this setting, combining a hierarchical architecture with strong inductive bias and a parameter count that grows only logarithmically with system size. Their appeal rests on the generalization bounds of Caro et al. (2022), which show that the generalization error of a quantum model scales with the number of trainable parameters rather than with the Hilbert-space dimension, placing QCNNs in a potentially sample-efficient regime. We develop a hardware-compatible QCNN with mid-circuit measurement and classical feed-forward, and show on a binary handwritten-digit task that strong test performance is achievable from as few as 10 training samples, with the generalization error decreasing as the training set grows. At a matched 45-parameter budget the QCNN learns where an equally small classical convolutional network stays at chance, although an unconstrained classical baseline with roughly 25,000 parameters remains strongest when data are plentiful. Transpiling amplitude and angle encoded circuits across image resolutions from 2x2 to 512x512 pixels then exposes the dominant scaling bottleneck: amplitude encoding stays qubit-efficient but grows extremely deep, whereas angle encoding stays shallow but becomes qubit-prohibitive. On the medically motivated BreastMNIST benchmark the QCNN does not surpass the unconstrained classical network, yet it learns consistently above chance using orders of magnitude fewer parameters. Our results indicate that for QCNNs, learning from few samples is attainable in practice, whereas scaling to realistic image data is constrained less by optimization than by data encoding and hardware execution.
Modern quantum machine learning (QML) methods involve variationally optimizing a parameterized quantum circuit on a training data set, and subsequently making predictions on a testing data set (i.e., generalizing). In this work, we provide a comprehensive study of generalization performance in QML after training on a limited number N of training data points. We show that the generalization error of a quantum machine learning model with T trainable gates scales at worst as [Formula: see text]. When only K ≪ T gates have undergone substantial change in the optimization process, we prove that the generalization error improves to [Formula: see text]. Our results imply that the compiling of unitaries into a polynomial number of native gates, a crucial application for the quantum computing industry that typically uses exponential-size training data, can be sped up significantly. We also show that classification of quantum states across a phase transition with a quantum convolutional neural network requires only a very small training data set. Other potential applications include learning quantum error correcting codes or quantum dynamical simulation. Our work injects new hope into the field of QML, as good generalization is guaranteed from few training data.
The ability to extract general laws from a few known examples depends on the complexity of the problem and on the amount of training data. In the quantum setting, the learner's generalization performance is further challenged by the destructive nature of quantum measurements that, together with the no-cloning theorem, limits the amount of information that can be extracted from each training sample. In this paper we focus on hybrid quantum learning techniques where classical machine-learning methods are paired with quantum algorithms and show that, in some settings, the uncertainty coming from a few measurement shots can be the dominant source of errors. We identify an instance of this possibly general issue by focusing on the classification of maximally entangled vs. separable states, showing that this toy problem becomes challenging for learners unaware of entanglement theory. Finally, we introduce an estimator based on classical shadows that performs better in the big data, few copy regime. Our results show that the naive application of classical machine-learning methods to the quantum setting is problematic, and that a better theoretical foundation of quantum learning is required.
Quantum machine learning is at the crossroads of two of the most exciting current areas of research: quantum computing and classical machine learning. Although the field is still in its infancy, the body of literature is already large enough to warrant several review articles [1–3]. This short survey focuses on a selection of significant recent results on the subtopic of quantum neural networks, an area that hopes to build on the enormous impact that classical neural networks have had. In particular, we concentrate on quantum generalizations of popular neural-network concepts such as Boltzmann machines, generative adversarial networks and autoencoders, as well as applications of classical neural networks to quantum information and vice versa. Examples of Boltzmann machines. In a fully connected Boltzmann machine, connections exist between all visible and hidden neurons. In a restricted Boltzmann machine (RBM), there are no connections between neurons of the same type. A deep Boltzmann machine (DBM) consists of adding one or more additional layers of hidden neurons to an RBM, with no connections between neurons in the same layer. The energy-based nature of BMs gives a natural framework for considering quantum generalizations of their behavior. The BM energy function is equivalent to the Hamiltonian of a simple Ising model and one might hope that more general Hamiltonians allowed by quantum mechanics could explain certain data sets better than classically. Amin et al. [4] adopt this approach, and take the so-called transverse field Ising Hamiltonian as the energy function. This increased generality brings certain difficulties, however, as the transverse field term makes training the quantum Boltzmann machine (QBM) non-trivial. Although the general scaling and performance of QBMs is not yet well understood, for certain small-size examples the authors observe that their QBMs can learn data distributions better than classical BMs. Kieferova and Wiebe [5] extend the formalism of Amin et al. and consider more general Hamiltonians that, unlike the transverse Ising model, are not in general believed to be classically efficiently simulatable. Furthermore, they propose new methods for training that allow for parameters corresponding to quantum terms to be learned. Finally, their QBM enables a form of quantum-state tomography that can learn non-classical data, and also generate copies of the reconstructed state. Again, small-scale numerical evidence suggests that their proposals may provide an advantage over classical BMs, although their tendency to overfit data must be investigated further. The practical value of BMs, either classical or quantum, relies in part on the ease with which they are trained. This in turn depends on sampling efficiently from the Boltzmann distribution. Classically this can be done approximately only for the special case of restricted Boltzmann machines (RBM), i.e. where connections are between visible and hidden units, and no connections exist between neurons of the same type. Wiebe et al. [6] propose two quantum algorithms for Boltzmann sampling, based on starting from a state corresponding to the product distribution that is closest to the target Boltzmann distribution. While small-scale numerical studies are encouraging, true bounds on the running time of their algorithms rely on a problem-dependent parameter κ, the scaling of which is hard to estimate. The hope remains though that empirically their algorithms work well in practice, even for multi-layer and fully connected BMs. The expressive power of BMs for compactly representing complicated probability distributions has led to another line of investigation, namely their use for representing quantum many-body wave functions. The authors apply this technique to two prototypical spin models and achieve high accuracy in both ground-state estimation and dynamical evolution, improving on existing state-of-the-art techniques. Their results indicate that RBM states can capture complex many-body correlations in an efficient manner—evidence reinforced by the work of Deng et al. [8], who show that certain topological quantum states in one, two and three dimensions can be represented by RBM states with a number of parameters only linear in the system size. In a subsequent study, however, Gao and Duan [9] prove that although RBM states can indeed represent many highly entangled states, their expressive power is limited in that they cannot represent arbitrary states generated by polynomial-sized quantum circuits, nor those that are the ground states of k-local Hamiltonians with polynomial-sized energy gaps, unless the polynomial hierarchy from computational complexity theory collapses. On the other hand, they show that these two important classes of states can be efficiently represented by a deep Boltzmann machine of three layers. Intriguingly though, another paper by Deng et al. [10] gives an analytic construction of a family of RBM states with a number of parameters linear in the system size that exhibits maximal volume-law entanglement, and that cannot be represented efficiently by either matrix product states (MPS) or tensor-network (TN) states. This shows that the expressive power of even efficiently parametrized RBM states is not limited by entanglement in the same way that other popular compact representations are. This gives the hope that RBM states may prove a valuable tool in understanding entangled quantum systems intractable by traditional methods. Levine et al. [11] establish another connection between the many-body wave function and neural networks, this time in the form of convolutional arithmetic circuits (ConvAC), a variant of the popular convolutional neural networks in widespread use today. By proving a one-to-one equivalence between a ConvAC circuit and TN quantum states, the authors are able to use tools from quantum information theory to analyze the entanglement present in the corresponding TN, and quantify the neural network's ability to model complex correlations between its inputs. This in turn can be used to obtain bounds on the expressivity of a given ConvAC architecture and be of practical use in choosing network parameters tailored to the data. While ConvAC are not widely used in practice, the authors empirically validate that the insights gained can be applied successfully to more common convolutional neural-network designs. Generative adversarial networks (GAN) are machine-learning algorithms in which two players—a generator and a discriminator—compete against each another. The generator's goal is to create and send fake data samples to the discriminator and maximize the probability that the discriminator erroneously believes that the sample comes from some true data distribution. The discriminator attempts to discern real from fake data as best as possible. Under reasonable assumptions the game converges to a point where the discriminator is unable to discriminate with probability greater than 1/2 (i.e. no better than random guessing), and the generator in fact has learned to generate data from the true distribution. Lloyd and Weedbrook [12] consider the natural extension in which one or both of the generator and discriminator are equipped with quantum information processors. When the data to be learned are a quantum state and both parties have quantum processors, the authors show that the unique end point of the adversarial game also coincides with the generator producing a quantum state corresponding to the true data statistics, and the discriminator unable to discern true from false data better than random guessing. They further argue that a different outcome may hold when the true samples are produced by the measurement of some quantum state, and the generator is purely classical. The concept of quantum supremacy is based on the assumption that there exist probability distributions accessible to a quantum computer that no classical computer can efficiently sample from. If the true data correspond to one of these distributions, then no classical generator can efficiently generate fake data reproducing the true statistics and, in principle, the discriminator can successfully discriminate true from fake data. However, whether the discriminator can efficiently find the measurements required to do so is unclear. Quantum-computing techniques have also found applications in autoencoders—neural networks where a larger number of input and output neurons are connected via a smaller number of intermediary latent neurons. The goal is to tune the weights of the network so that a training set of classical bit strings can be approximately recovered after being passed through the autoencoder. Romero et al. [13] consider instead a training set of quantum states, and ask whether a quantum circuit with a similar bottleneck structure can compress and approximately recover the data. The circuits that they consider consist of gates chosen from some set of parametrized unitary operations, and arranged in some fixed order. The parameters are then optimized classically so that the circuit best predicts the training-set labels. The authors demonstrate the validity of this approach by showing that ground states of the Hubbard model and certain molecules can indeed be compressed and recovered by such circuits. However, in general, the performance of such quantum autoencoders will be highly dependent on the kind of programmable circuit on which they are based, and at present there does not seem to be a structured way of determining this. Quantum neural networks are still in their infancy and, while the above works are important steps towards bridging our understanding between classical neural networks and quantum computing, many questions remain unanswered. If the expressivity of Boltzmann machine states is not restricted by entanglement in the same way that MPS and tensor-network states are, what is the best way of understanding their limitations? How can physicists use the correspondence between ConvAC and TN to their advantage in finding efficient, expressive representations of many-body wave functions? Can classical data distributions be learned more efficiently by quantum GAN than classical GAN? Whatever the outcome of these and many other related questions, this promises to be an exciting and active area of research in the years to come.