Predictive Performance of Deep Quantum Data Re-uploading Models
Xin Wang, Han-Xiao Tao, Re-Bing Wu
TL;DR
The paper proves a fundamental limit on the predictive performance of deep data re-uploading quantum models with limited qubits: as encoding depth $L$ increases, the expected output over unseen data converges toward the maximally mixed state, causing near-random predictions. By analyzing Pauli-basis coefficients and employing the Petz-Rényi-2 divergence $D_2$, the authors derive a bound $D_2(\mathbb{E}[\rho]\|\rho_I) \le \log_2\bigl(1+(2^N-1)e^{-L\sigma^2}\bigr)$, and show that for $L$ large enough, $|\mathbb{E}_{\boldsymbol{x}}[h_S(\boldsymbol{x})]-h_I|\le \epsilon$, with $h_I=\operatorname{Tr}[H\rho_I]$. They extend these results to arbitrary parameterized gates and to repeated data uploading, proving that increasing depth, not repetitions, governs predictive degradation. Experiments on synthetic and real datasets (e.g., MNIST, CIFAR-10) corroborate the theory, revealing that deep encoding layers on few-qubit circuits yield predictions near random-guessing, while training error can improve with more layers or repetitions but generalization remains poor. The practical implication is clear: for high-dimensional classical data, quantum classifiers should prioritize wider circuit architectures over deeper encodings to retain predictive power.
Abstract
Quantum machine learning models incorporating data re-uploading circuits have garnered significant attention due to their exceptional expressivity and trainability. However, their ability to generate accurate predictions on unseen data, referred to as the predictive performance, remains insufficiently investigated. This study reveals a fundamental limitation in predictive performance when deep encoding layers are employed within the data re-uploading model. Concretely, we theoretically demonstrate that when processing high-dimensional data with limited-qubit data re-uploading models, their predictive performance progressively degenerates to near random-guessing levels as the number of encoding layers increases. In this context, the repeated data uploading cannot mitigate the performance degradation. These findings are validated through experiments on both synthetic linearly separable datasets and real-world datasets. Our results demonstrate that when processing high-dimensional data, the quantum data re-uploading models should be designed with wider circuit architectures rather than deeper and narrower ones.