Complexity phase transitions in instantaneous quantum polynomial-time circuits

We study a subclass of the Instantaneous Quantum Polynomial-time (IQP) circuit with a varying density of two-qubit gates. We identify two phase transitions as a function of the gate density. At the first transition, the coherent Gibbs representation of the output state transitions from quasilocal to extended, and becomes completely non-local after the second transition, resembling that of Haar random states. We introduce several new diagnostics to argue that learning the distribution after the first transition already becomes classically intractable, even though its output distribution is far from random. Our work thus opens a novel pathway for exploring quantum advantage in more practical setups whose outcome distributions are more structured than random states.