| Summary: | Abstract Ensuring data privacy in machine learning models is critical, especially in distributed settings where model gradients are shared among multiple parties for collaborative learning. Motivated by the increasing success of recovering input data from the gradients of classical models, this study investigates the analogous challenge for variational quantum circuits (VQC) as quantum machine learning models. We highlight the crucial role of the dynamical Lie algebra (DLA) in determining privacy vulnerabilities. While the DLA has been linked to the trainability and simulatability of VQC models, we establish its connection to privacy for the first time. We show that properties conducive to VQC trainability, such as a polynomial-sized DLA, also facilitate extracting detailed snapshots of the input, posing a weak privacy breach. We further investigate conditions for a strong privacy breach, where original input data can be recovered from snapshots by classical or quantum-assisted methods. We establish properties of the encoding map, such as classical simulatability, overlap with DLA basis, and its Fourier frequency characteristics that enable such a privacy breach of VQC models. Our framework thus guides the design of quantum machine learning models, balancing trainability and robust privacy protection.
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