Generalization bounds for quantum machine learning

Short answer: Generalization bounds say how far a quantum model's error on new data can exceed its error on training data. A 2022 Nature Communications paper proves that for a model with T trainable gates trained on N data points, the generalization error scales at worst as sqrt(T/N), improving to sqrt(K/N) when only K of the T gates change substantially. Other papers add bounds that depend on data encoding and on information-theoretic quantities. These are theory results, not demonstrations on hardware.

Trainable gates and training data

The 2022 paper studies variational quantum machine learning: optimise a parameterised quantum circuit on a training set, then predict on a test set. The authors show that the generalization error of a model with T trainable gates scales at worst as sqrt(T/N) after training on N data points. When only K gates, with K much smaller than T, undergo substantial change in the optimisation, they prove the error improves to sqrt(K/N). The authors say this implies that compiling unitaries into a polynomial number of native gates, which they call a crucial application for the quantum computing industry that typically uses exponential-size training data, can be sped up significantly. They also report that classification of quantum states across a phase transition with a quantum convolutional neural network requires only a very small training data set.

Data encoding matters

A 2021 paper in Quantum notes that theoretical guarantees on the out-of-sample performance of parametrised quantum circuit (PQC) models had emerged, but that none depended explicitly on how classical input data is encoded into the circuit. The authors derive bounds that do depend on the encoding strategy. They obtain them by bounding the complexity of PQC-based models using Rademacher complexity and metric entropy, two complexity measures from statistical learning theory. They say the results facilitate choosing data-encoding strategies through structural risk minimisation, a framework for model selection.

Learning from quantum data

A 2024 paper in the Proceedings of Machine Learning Research starts from the observation that many directions of quantum learning theory, from state discrimination and metrology to quantum probably approximately correct (PAC) learning and shadow variants of state tomography, had evolved separately. The authors propose a formalism in which a learner trains on classical-quantum data and is tested on new data. They prove bounds on the expected generalization error in terms of classical and quantum information-theoretic quantities that measure how strongly the learner's hypothesis depends on the training data. They use tools from quantum optimal transport and quantum concentration inequalities.

What the bounds do not show

A bound limits the gap between training and test error. It does not say the training error is low, that the optimisation will find a good solution, or that a quantum model beats a classical one. The 2022 paper's scaling is an upper bound on the error, stated as holding at worst. None of the three papers reports a hardware experiment on large data sets, and the application areas they mention remain at the research stage.

FAQ

What does a generalization bound measure?

The gap between a model's performance on training data and on unseen data.

What is the scaling for trainable gates?

The 2022 paper proves error scaling at worst as sqrt(T/N) for T trainable gates and N training points, and sqrt(K/N) when only K gates change substantially.

Does the way data is encoded matter?

The 2021 paper derives bounds that depend explicitly on the data-encoding strategy, which earlier bounds did not.

Does a bound guarantee a good model?

No. It limits the generalization gap only. It does not guarantee low training error or an advantage over classical methods.

Is this tested on hardware?

These are theory papers. They do not report large hardware experiments.