  We study random constant-depth quantum circuits in a two-dimensional
architecture. While these circuits only produce entanglement between nearby
qubits on the lattice, long-range entanglement can be generated by measuring a
subset of the qubits of the output state. It is conjectured that this
long-range measurement-induced entanglement (MIE) proliferates when the circuit
depth is at least a constant critical value. For circuits composed of
Haar-random two-qubit gates, it is also believed that this coincides with a
quantum advantage phase transition in the classical hardness of sampling from
the output distribution. Here we provide evidence for a quantum advantage phase
transition in the setting of random Clifford circuits. Our work extends the
scope of recent separations between the computational power of constant-depth
quantum and classical circuits, demonstrating that this kind of advantage is
present in canonical random circuit sampling tasks. In particular, we show that
in any architecture of random shallow Clifford circuits, the presence of
long-range MIE gives rise to an unconditional quantum advantage. In contrast,
any depth-d 2D quantum circuit that satisfies a short-range MIE property can be
classically simulated efficiently and with depth O(d). Finally, we introduce a
two-dimensional, depth-2, "coarse-grained" circuit architecture, composed of
random Clifford gates acting on O(log n) qubits, for which we prove the
existence of long-range MIE and establish an unconditional quantum advantage.
