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We study a subclass of the Instantaneous Quantum Polynomial-time (IQP)circuit with a varying density of two-qubit gates. In addition to a knownanticoncentration regime, we identify novel parameter conditions where themodel is classically simulable or the output distribution follows thePorter-Thomas distribution. By showing that those parameter regimes do notcoincide, we argue the presence of more than two phases in the model. Thelearnability of the output distribution of this model is further studied, whichindicates that an energy-based model fails to learn the output distributioneven when it is not anticoncentrated. Our study reveals that a quantum circuitmodel can have multiple fine-grained complexity phases, suggesting thepotential for quantum advantage even when the output distribution is far fromthe Porter-Thomas distribution.

Complexity phase transitions in instantaneous quantum polynomial-time circuits