Fig. 1: Schematic representation of our results.

Many works in the literature have shown that BPs can arise from: the expressiveness of the parametrized circuit, the locality of the measurement operator, the entanglement in the initial state, or due to hardware noise. In the figure, we schematically depict these sources as being separate, since their study is usually performed in restricted scenarios, thus limiting our ability to interconnect them. In this work, we present a unified Lie algebraic theory that can be used to study BPs in deep parametrized quantum circuits under SPAM and coherent noise. Our theorems allow us to understand under a single unified framework all sources of BPs by leveraging concepts such as generalized locality entanglement and algebraic decoherence.