Multifractality in non-unitary random dynamics

We explore the multifractality of the steady state wave function in non-unitary random quantum dynamics in one dimension. We focus on two classes of random systems: the hybrid Clifford circuit model and the non-unitary free fermion dynamics. In the hybrid Clifford model, we map the measurement driven transition to an Anderson localization transition in an effective graph space by using properties of the stabilizer state. We show that the volume law phase with nonzero measurement rate is non-ergodic in the graph space and exhibits weak multifractal behavior. We apply the same method to the hybrid Clifford quantum automaton circuit and obtain similar multifractality in the volume law phase. For the non-unitary random free fermion system with a critical steady state, we compute the moments of the probability distribution of the single particle wave function and demonstrate that it is also weakly multifractal and has strong variations in real space.