Integrable nonunitary open quantum circuits

We explicitly construct an integrable interacting dissipative quantum circuit, via a trotterization of the Hubbard model with imaginary interaction strength. To prove integrability, we build an inhomogeneous transfer matrix, from which conserved super-operator charges can be derived, in particular, the circuit's dynamical generator. After showing the trace preservation and complete positivity of local maps, we reinterpret them as the Kraus representation of the dynamics of free fermions with single-site dephasing. The integrability of the map is broken by adding interactions to the local coherent dynamics or by removing the dephasing. Moreover, the construction allows us to explicitly build circuits belonging to different non-Hermitian symmetry classes, which are characterized by the behavior under transposition instead of complex conjugation. We confirm all our analytical results by using complex spacing ratios to examine the spectral statistics of the dissipative circuits.