  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.
