Influence-solvability: a systematic theory of (1+1)D solvability and its application to brickwork circuits
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 χ. 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 χ=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.
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BibTeX
@misc{hubner2026influencesolvability,
title = {{Influence-solvability: a systematic theory of $(1+1)D$ solvability and its application to brickwork circuits}},
author = {Friedrich Hübner and Sun Woo P. Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.12538},
eprint = {2606.12538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12538},
}
RIS
TY - UNPB
TI - Influence-solvability: a systematic theory of (1+1)D solvability and its application to brickwork circuits
AU - Friedrich Hübner
AU - Sun Woo P. Kim
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2606.12538
UR - https://arxiv.org/abs/2606.12538
AB - `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 χ. 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 χ=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.
ER -