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Encoding dependent generalization bounds for parametrized quantum circuits

Caro, M.C., Gil Fuster, E., Meyer, J.J., Eisert, J., Sweke, R.

Published Jan 1, 2021• 2 citations
OpenAlex

Abstract

A large body of recent work has begun to explore the potential of parametrized quantum circuits PQCs as machine learning models, within the framework of hybrid quantum classical optimization. In particular, theoretical guarantees on the out of sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC based models that depend explicitly on the strategy used for data encoding. These imply bounds on the performance of trained PQC based models on unseen data. Moreover, our results facilitate the selection of optimal data encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC based models via trigonometric functions. Our generalization bounds emphasize the importance of well considered data encoding strategies for PQC based models

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A large body of recent work has begun to explore the potential of parametrized quantum circuits PQCs as machine learning models, within the framework of hybrid quantum classical optimization. In particular, theoretical guarantees on the out of sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC based models that depend explicitly on the strategy used for data encoding. These imply bounds on the performance of trained PQC based models on unseen data. Moreover, our results facilitate the selection of optimal data encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC based models via trigonometric functions. Our generalization bounds emphasize the importance of well considered data encoding strategies for PQC based models
A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.
A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.
Understanding generalization is essential for assessing the practical viability of quantum machine learning. In many applications, quantum models operate on classical data that must first be encoded into quantum states, making the encoding part of the effective hypothesis class. We develop a margin-based generalization framework for this setting, covering both fixed and trainable quantum data encodings. For fixed encodings, the existing margin bound transfers unchanged. For trainable encodings, we derive a bound using a two-scale covering argument that combines the original margin-dependent term with an additive encoding-capacity term of order O~(L/m)\tilde O(\sqrt{L/m})O~(L/m​), where LLL is the number of trainable encoding parameters and mmm is the training-sample size. Across 19,440 simulated training runs spanning various image datasets, circuit architectures, and optimization settings, the inverse margin performs best among five baseline and two bound-derived complexity measures under conditional-independence and 90th-percentile sign-error criteria, within bootstrap uncertainty. These results provide a unified margin-based view of generalization in quantum machine learning with classical data.
In the field of quantum machine learning (QML), parametrized quantum circuits (PQCs)─constructed using a combination of fixed and tunable quantum gates─provide a promising hybrid framework for tackling complex machine learning problems. Despite numerous proposed applications, there remains limited exploration of data sets relevant to quantum chemistry. In this study, we investigate the potential benefits and limitations of PQCs on two chemically meaningful data sets: (1) the BSE49 data set, containing bond separation energies for 49 different classes of chemical bonds, and (2) a data set of water conformers, where coupled-cluster singles and doubles (CCSD) wave functions are predicted from lower-level electronic structure methods using the data-driven coupled-cluster (DDCC) approach. We construct a comprehensive set of 168 PQCs by combining 14 data encoding strategies with 12 variational ansätze, and evaluate their performance on circuits with 5 and 16 qubits. Our initial analysis examines the impact of circuit structure on model performance using state-vector simulations. We then explore how circuit depth and training set size influence model performance. Finally, we assess the performance of the best-performing PQCs on current quantum hardware, using both noisy simulations ("fake" backends) and real quantum devices. Our findings underscore the challenges of applying PQCs to chemically relevant problems that are straightforward for classical machine learning methods but remain nontrivial for quantum approaches.
In this study, we analyze the Rademacher complexity $ \mathcal{R}_{M} $ of a parameterized unitary whose generators are chosen from $ n $-qubit Pauli strings. Although generalization bounds for quantum machine learning models have been studied in several settings, explicit Rademacher-complexity bounds for parameterized unitaries generated by Pauli strings remain less transparent. We derive simple scaling bounds in terms of the number of parameters $ L $ and the number of training samples $ M $: $ \mathcal{O}(\frac{L^{\frac{3}{2}}}{\sqrt{M}}) $ for the full parameter domain and $ \mathcal{O}(\frac{L}{\sqrt{M}}) $ for a restricted parameter domain. Furthermore, we compare the obtained results with those for a classical linear model class and suggest a potential statistical-complexity advantage when the norms of both the input and the parameter in the classical model scale with the number of parameters. Numerical experiments provide qualitative evidence consistent with the predicted scaling.
Estimating quantum entropies and divergences is an important problem in quantum physics, information theory, and machine learning. Quantum neural estimators (QNEs), which utilize a hybrid classical-quantum architecture, have recently emerged as an appealing computational framework for estimating these measures. Such estimators combine classical neural networks with parametrized quantum circuits, and their deployment typically entails tedious tuning of hyperparameters controlling the sample size, network architecture, and circuit topology. This work initiates the study of formal guarantees for QNEs of measured (Rényi) relative entropies in the form of non-asymptotic error risk bounds. We further establish exponential tail bounds showing that the error is sub-Gaussian and thus sharply concentrates about the ground truth value. For an appropriate sub-class of density operator pairs on a space of dimension ddd with bounded Thompson metric, our theory establishes a copy complexity of O(∣Θ(U)∣d/ε2)O(|Θ(\mathcal{U})|d/ε^2)O(∣Θ(U)∣d/ε2) for QNE with a quantum circuit parameter set Θ(U)Θ(\mathcal{U})Θ(U), which has minimax optimal dependence on the accuracy εεε. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to O(∣Θ(U)∣polylog(d)/ε2)O(|Θ(\mathcal{U})|\mathrm{polylog}(d)/ε^2)O(∣Θ(U)∣polylog(d)/ε2). Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.