  Differential privacy provides a theoretical framework for processing a
dataset about $n$ users, in a way that the output reveals a minimal information
about any single user. Such notion of privacy is usually ensured by
noise-adding mechanisms and amplified by several processes, including
subsampling, shuffling, iteration, mixing and diffusion. In this work, we
provide privacy amplification bounds for quantum and quantum-inspired
algorithms. In particular, we show for the first time, that algorithms running
on quantum encoding of a classical dataset or the outcomes of quantum-inspired
classical sampling, amplify differential privacy. Moreover, we prove that a
quantum version of differential privacy is amplified by the composition of
quantum channels, provided that they satisfy some mixing conditions.
