New research has demonstrated that training a quantum neural network requires only a small amount of data. The discovery, “Generalization in quantum machine learning from few training data,” led by Patrick Coles of the Theoretical Division, Los Alamos National Laboratory, USA, challenges earlier assumptions based on conventional computing’s appetite for large data in machine learning or artificial intelligence. As published in Nature Communications on August 22, 2022, the theorem has immediate applications, including faster compilation for quantum computers and distinguishing phases of matter for material discovery.
“Many people believe that quantum machine learning will require a lot of data. We have rigorously shown that for many relevant problems, this is not the case,” “This provides new hope for quantum machine learning. We’re closing the gap between what we have today and what’s needed for quantum advantage when quantum computers outperform classical computers.”
Lukasz Cincio, a quantum theorist at Los Alamos National Laboratory and co-author of the paper containing the proof published in the journal Nature Communications.
Quantum Machine Learning (QML)
Modern Quantum Machine Learning (QML) methods entail variationally optimizing a parameterized quantum circuit on a training data set, followed by predictions on a testing data set (i.e., generalizing). In this research, Patrick Coles and his team present a comprehensive study of generalization performance in QML after training on a limited number N of training data points.
Machine Learning (ML) aims to make precise predictions on previously unseen data; this is known as generalization. Consequently, considerable effort has gone into understanding the generalization capabilities of classical ML models.
“The need for large data sets could have been a roadblock to quantum AI, but our work removes this roadblock. While other issues for quantum AI could still exist, at least now we know that the size of the data set is not an issue,”
Patrick Coles, a quantum theorist at the Laboratory and co-author of the paper.
In modern QML, training a parameterized quantum circuit to analyze either classical or quantum data sets is typical. Early results suggest that, in certain circumstances, QML models may outperform classical models for classical data analysis. In addition, it has been demonstrated that QML models can provide an exponential advantage in sample complexity when analyzing quantum data. Little is known, however, about the conditions required for accurate generalization in QML.
Neural Networks and Quantum Algorithms
All AI systems need data to train neural networks to recognize or generalize previously unseen data in real-world applications. The size of a mathematical construct known as a Hilbert space, which becomes exponentially large for training over large numbers of qubits, is assumed to determine the number of parameters or variables. Significant progress has been made in understanding QML model trainability, but trainability is different from generalization. Overfitting training data can be an issue for QML, as it is for traditional machine learning. Furthermore, the training data size required for QML generalization is yet to be investigated.
You may expect exponential training points when training a function acting on an exponentially sizeable Hilbert space. For example, research has shown that, exponentially in n (number of qubits), extensive training data would be required to train an arbitrary unitary. This implies exponential scaling of the resources needed for QML, which the field of quantum computation would like to avoid.
This research establishes general theoretical bounds on the generalization error in variational QML. First, they cover several limitations for the class of quantum operations that a variational QML model can implement. Then use the chaining method for random processes to derive generalization error bounds from them.
One important implication of their results is that a QML model is easily implementable and only requires a small quantity of training data to achieve high generalization. Their theorem also demonstrates that generalization increases when only a few parameters have changed significantly during the optimization process.
As a result, even if they used more parameters than the training data size, the QML model could still generalize effectively if some parameters changed. This implies that QML researchers should be careful not to overtrain their models, especially when the decrease in training error is minimal.
Practical Applications of Quantum Algorithms: QML
A classic application of QML is quantum phase classification, to which QCNNs have been effectively applied. However, previous works presented a heuristic explanation for QCNNs’ strong generalization performance.
In this paper, the researchers have formulated a theory that describes the behaviour of QCNNs and have confirmed it numerically for a wide range of system sizes. Their study will allow scientists to go beyond the particular QCNN model to derive broad guidelines for ensuring successful generalization.
For an experimenter to generate training data for a QCNN problem, he will require different phases of matter and states, which necessitates careful tuning of other parameters in the underlying Hamiltonian. Therefore, good generalization with small training data sizes is critical for implementing phase classification through QML in physical experiments.
Several practical unitary compilation algorithms make use of training data. In earlier works, the number of training data sets increased exponentially with the number of qubits. This scaling implies a similarly poor scaling of the computational complexity of processing the data. Also, in physical implementations, generating training data might be expensive. Theoretically, the generalization bounds from this work offer assurances regarding the performance of unitary compilation with just polynomial-size training data.
An example of an optimization problem is finding quantum error-correcting codes. And because estimating the average fidelity of the code requires training data, it may be seen as a machine learning task (e.g., chosen from a 2-design). On traditional computers, much effort has been made to tackle this problem. As this research applies, the generalization bounds can benefit such techniques, leading to speedier classical quantum code discovery.
Quantum autoencoders and Generative Adversarial Networks (GANs) have recently been generalized to the quantum setting. Because both use training data, these generalization bounds give quantitative guidelines on how much training data to use in these applications.
Furthermore, their results may be used to guide ansatz design. While no standard ansatz exists for quantum autoencoders or quantum GANs, ansätze with the fewest parameters will likely result in the highest generalization performance.
