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