Integrable nonunitary quantum circuits

We show that the integrable Lindblad superoperators found recently can be used to build nonunitary quantum circuits with two-site gates by demonstrating $\check{R}$-matrices are completely positive and trace preserving. Using bond-site transformation, we obtain corresponding three-site gates. When restricted diagonals of density matrix, these models reduce classical cellular automata. This approach may pave way for a systematic construction circuits.