Observing the measurement-induced transition without exponential postselection
In plain words
Seeing the transition directly requires repeating the same random set of measurement outcomes, which happens with exponentially small probability. A way to detect it in generic circuits with resources that grow only polynomially with size is not established.
Precise statement
For non-Clifford monitored circuits on $N$ qubits under Born-rule measurements, find an estimator that detects $p_c$ with $\operatorname{poly}(N)$ samples and $\operatorname{poly}(N)$ classical processing, or prove that every such probe needs $\operatorname{exp}(N)$ resources. Known probes use Clifford circuits, require classically simulating the circuit (cross-entropy benchmark of Li, Zou, Glorioso, Altman and Fisher, PRL 130, 220404, 2023), or postselect.
What would settle it
An efficient detection protocol with proof and a demonstration on hardware, or a lower bound on sample complexity.
Status in the literature
Google Quantum AI (Nature 2023) observed measurement-induced entanglement signatures on a processor of about 70 qubits (approximate) using decoding methods; the general question is open.