Friedrich Hübner: MPS equivalence and solvability for brickwork circuits

Solvable circuits, such a 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 the bath can be written as a translation invariant MPS). The result is based on a similar local condition for when two MPS of different bond dimensions are equivalent.

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.