Influence-solvability: a systematic theory of $(1+1)D$ solvability and its application to brickwork circuits

arXiv:2606.12538v2 Announce Type: replace-cross Abstract: `Solvable' circuits, such as dual unitaries and its generalisations, have arisen as paradigmatic examples of tractable chaotic non-equilibrium dynamics, both in classical and quantum systems. However, while increasingly complicated sufficient conditions have been proposed, a systematic theory classifying and understanding general features of Solvable circuits is missing. In this work we develop such a theory by deriving the most general local necessary and sufficient conditions under which a $(1+1)D$ circuit is Solvable. Concretely, we define a circuit to be \emph{influence-solvable} if its influence matrix, representing the `bath' generated by its own evolution, is given by a uniform MPS with finite bond-dimension $\chi$. This property allows for efficient computation of subsystem dynamics and essentially contains all known examples of Solvable circuits. The set of necessary and sufficient local conditions follows from a new version of the fundamental theorem of MPS for open boundary conditions. Next, we apply our theory to brickwork circuits with $\chi=1$ influence-solvability and perform a systematic classification of classical brickwork circuits with local dimension up to $d=3$ and quantum brickwork circuits with $d=2$. Our search reveals new Solvable circuits that are not captured by known conditions.

Abstract:`Solvable' circuits, such as dual unitaries and its generalisations, have arisen as paradigmatic examples of tractable chaotic non-equilibrium dynamics, both in classical and quantum systems. However, while increasingly complicated sufficient conditions have been proposed, a systematic theory classifying and understanding general features of Solvable circuits is missing. In this work we develop such a theory by deriving the most general local necessary and sufficient conditions under which a $(1+1)D$ circuit is Solvable. Concretely, we define a circuit to be \emph{influence-solvable} if its influence matrix, representing the `bath' generated by its own evolution, is given by a uniform MPS with finite bond-dimension $\chi$. This property allows for efficient computation of subsystem dynamics and essentially contains all known examples of Solvable circuits. The set of necessary and sufficient local conditions follows from a new version of the fundamental theorem of MPS for open boundary conditions. Next, we apply our theory to brickwork circuits with $\chi=1$ influence-solvability and perform a systematic classification of classical brickwork circuits with local dimension up to $d=3$ and quantum brickwork circuits with $d=2$. Our search reveals new Solvable circuits that are not captured by known conditions.

Statistical Mechanics (cond-mat.stat-mech); Quantum Physics (quant-ph)

arXiv:2606.12538 [cond-mat.stat-mech]

(or arXiv:2606.12538v2 [cond-mat.stat-mech] for this version)

https://doi.org/10.48550/arXiv.2606.12538 Focus to learn more arXiv-issued DOI via DataCite

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Influence-solvability: a systematic theory of $(1+1)D$ solvability and its application to brickwork circuits

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