  We investigate the prospects of employing the linear cross-entropy to
experimentally access measurement-induced phase transitions (MIPT) without
requiring any postselection of quantum trajectories. For two random circuits
that are identical in the bulk but with different initial states, the linear
cross-entropy $\chi$ between the bulk measurement outcome distributions in the
two circuits acts as a boundary order parameter, and can be used to distinguish
the volume law from area law phases. In the volume law phase (and in the
thermodynamic limit) the bulk measurements cannot distinguish between the two
different initial states, and $\chi = 1$. In the area law phase $\chi < 1$. For
circuits with Clifford gates, we provide numerical evidence that $\chi$ can be
sampled to accuracy $\epsilon$ from $O(1/\epsilon^2)$ trajectories, by running
the first circuit on a quantum simulator without postselection, aided by a
classical simulation of the second. We also find that for weak depolarizing
noise the signature of the MIPT is still present for intermediate system sizes.
In our protocol we have the freedom of choosing initial states such that the
"classical" side can be simulated efficiently, while simulating the "quantum"
side is still classically hard.
