Universality class of the measurement-induced transition in Haar circuits
In plain words
At the transition, quantities like entanglement follow power laws with universal exponents, as at a boiling point. These exponents and the theory that produces them are not known for ordinary random circuits of qubits.
Precise statement
For $1+1D$ brickwork circuits of Haar-random two-qudit gates (local dimension q) with projective measurements at rate p, determine the critical exponents at $p_c$ (correlation-length exponent nu, entanglement scaling and surface exponents) and identify the field theory, for finite q and as q goes to $\infty$ where the transition maps to classical percolation. Qubit numerics give $\nu \sim 1.2-1.3$ (approximate) with other exponents distinct from percolation (Zabalo et al. 2020).
What would settle it
An analytic theory (e.g. the replica limit of the associated statistical model) giving exponents confirmed by large-scale numerics.