Circuits of space and time quantum channels

Exact solutions in interacting many -body systems are scarce but extremely valuable since they provide insights into the dynamics. Dual-unitary models are ex-amples in one spatial dimension where this is possible. These brick-wall quantum cir-cuits consist of local gates, which remain unitary not only in time, but also when interpreted as evolutions along the spatial directions. However, this setting of uni-tary dynamics does not directly apply to real-world systems due to their imperfect isolation, and it is thus imperative to con-sider the impact of noise to dual-unitary dynamics and its exact solvability. In this work we generalise the ideas of dual-unitarity to obtain exact solutions in noisy quantum circuits, where each uni-tary gate is substituted by a local quan-tum channel. Exact solutions are ob-tained by demanding that the noisy gates yield a valid quantum channel not only in time, but also when interpreted as evolu-tions along one or both of the spatial di-rections and possibly backwards in time. This gives rise to new families of mod-els that satisfy different combinations of unitality constraints along the space and time directions. We provide exact solu-tions for the spatio-temp oral correlation functions, spatial correlations after a quan-tum quench, and the structure of steady states for these families of models. We show that noise unbiased around the dual -unitary family leads to exactly solvable models, even if dual-unitarity is strongly violated. We prove that any channel uni-tal in both space and time directions can be written as an affine combination of a particular class of dual-unitary gates. Fi-nally, we extend the definition of solvable initial states to matrix-product density op-erators. We completely classify them when their tensor admits a local purification. particular class of dual-unitary gates. Fi-nally, we extend the definition of solvable initial states to matrix-product density op-erators. We completely classify them when their tensor admits a local purification.