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Description
Solvable circuits, such as dual unitary circuits and their extensions, have arisen as paradigmatic examples of tractable chaotic non-equilibrium dynamics, both in classical and quantum systems. These correspond to local algebraic relations which allow for calculation of observables due to a simplification of the corresponding tensor network. However, so far these relations are not exhaustive, and it is not clear what their limitations are. We fill this gap by providing a sufficient and necessary local condition under which a circuit is solvable (by this we mean that its influence matrix can be written as a translation invariant MPS). The result is based on a version of the fundamental theorem of MPS with open boundary conditions. We then apply these conditions to study the simplest case: factorized initial state and Markovian bath. For this case we classify all solvable classical circuits (with local dimension 2 and 3) and all solvable quantum circuits with local dimension 2.