Statistical Models of Barren Plateaus and Anti-Concentration of Pauli Observables

We introduce statistical models for each of the three main sources of barren plateaus: nonlocality of the observable, entanglement of the initial state, and circuit expressivity. For instance, non-local observables are modeled by random Pauli operators, which lead to barren plateaus with probability exponentially close to one. These models are complementary to the conventional deterministic ones, and often simpler to analyze. Using this framework, we show that in the barren plateau regime any two Pauli observables are anti-concentrated with high probability in the following sense. While each of the observables is localized in an exponentially small parameter subspace, these regions are essentially independent, so that their overlap is yet exponentially smaller than each subspace. This invites to rethink the structure of quantum landscapes with barren plateaus and approaches to their optimization, including warm-start strategies.