Exact solutions in interacting many-body systems are scarce but extremely valuable since they provide insights into the dynamics. Dual-unitary models are examples in one spatial dimension where this is possible. These brick-wall quantum circuits 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 unitary dynamics does not directly apply to real-world systems due to their imperfect isolation, and it is thus imperative to consider 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 unitary gate is substituted by a local quantum channel. Exact solutions are obtained by demanding that the noisy gates yield a valid quantum channel not only in time, but also when interpreted as evolutions along one or both of the spatial directions and possibly backwards in time. This gives rise to new families of models that satisfy different combinations of unitality constraints along the space and time directions. We provide exact solutions for the spatio-temporal correlation functions, spatial correlations after a quantum 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 unital in both space and time directions can be written as an affine combination of a particular class of dual-unitary gates. Finally, we extend the definition of solvable initial states to matrix-product density operators. We completely classify them when their tensor admits a local purification.
In our work, we generalize one such example –dual-unitary circuits– to systems beyond unitary dynamics, called space-time channels. Here coupling with the environment results in quantum dynamics consisting of local quantum channels, that is, open-system evolution. These space-time quantum channels are characterized by the property that the evolution is still physical upon changing the roles of space and time, exactly as in the case of dual-unitary circuits. This property defines different rich families of models with tractable dynamics.
Our work opens new doors to exactly solvable open quantum circuits. As quantum evolution, simulation, or computation is never entirely isolated from the environment this knowledge is much needed. Moreover, our work also explains why the signature of dual-unitarity (vanishing correlations inside the light cone), which was already witnessed in the experiment, is preserved under typical noise.
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