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

• URL: http://arxiv.org/abs/2606.12538v1
• Date: Wed, 10 Jun 2026 18:00:08 GMT
• Title: Influence-solvability: a systematic theory of $(1+1)D$ solvability and its application to brickwork circuits
• Abstract summary: We introduce influence-solvable circuits, a class of $(1+1)D$ circuits whose influence matrix is given by a uniform MPS with finite bond-dimension $$.<n>We perform a systematic classification of classical brickwork circuits with local dimension up to $d=3$ and quantum brickwork circuits with $d=2$.<n>Our search reveals new solvable circuits that are not captured by known solvability conditions.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 more complicated sufficient conditions have been proposed, a systematic theory classifying and understanding general features of solvable circuits is missing. We develop such a theory by introducing influence-solvable circuits, a class of $(1+1)D$ circuits whose influence matrix, which represents 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. We derive a set of necessary and sufficient local conditions by using a 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 solvability conditions.

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