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arxiv: 2606.12538 · v1 · pith:AHS5L2JLnew · submitted 2026-06-10 · ❄️ cond-mat.stat-mech · quant-ph

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

Friedrich H\"ubner , Sun Woo P. Kim This is my paper

Pith reviewed 2026-06-27 07:53 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords influence-solvabilitysolvable circuitsbrickwork circuitsmatrix product statesdual unitariesnon-equilibrium dynamicschaotic circuitslocal conditions
0
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The pith

Influence-solvable circuits unify known solvable dynamics by representing their generated bath as a uniform matrix product state with finite bond dimension and supply local conditions plus new examples in brickwork circuits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines influence-solvable circuits as those whose influence matrix, which encodes the effective bath produced by the circuit's own evolution, takes the form of a uniform matrix product state with finite bond dimension χ. This representation permits efficient calculation of subsystem dynamics and is shown to include essentially every previously identified solvable circuit. Using the fundamental theorem of matrix product states for open boundaries, the authors obtain necessary and sufficient local conditions on the circuit gates. The same framework is then used to classify all classical brickwork circuits with local dimension at most 3 and all quantum brickwork circuits with local dimension 2, uncovering solvable gates outside earlier sufficient criteria.

Core claim

Influence-solvable circuits are those whose influence matrix, representing the bath generated by the circuit 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. Using a version of the fundamental theorem of MPS for open boundary conditions, the paper derives a set of necessary and sufficient local conditions for influence-solvability. When applied to brickwork circuits, this yields a systematic classification of classical brickwork circuits with local dimension up to d=3 and quantum brickwork circuits with d=2, revealing new solvable circuits n

What carries the argument

The influence matrix represented as a uniform matrix product state with finite bond dimension χ, which encodes the bath generated by the circuit and permits application of the fundamental theorem of MPS to obtain local solvability conditions.

If this is right

  • Subsystem dynamics become efficiently computable once the influence matrix has finite bond dimension.
  • All previously known solvable circuits, including dual-unitary ones, satisfy the influence-solvability condition.
  • Necessary and sufficient local gate conditions replace collections of merely sufficient criteria.
  • New solvable brickwork circuits exist beyond those found by earlier methods, for classical local dimensions up to 3 and quantum local dimension 2.
  • A complete classification is obtained for the listed small local dimensions in the brickwork geometry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same influence-matrix construction could be applied to circuit geometries other than brickwork to search for additional solvable families.
  • The newly identified circuits may display distinctive correlation or entanglement patterns that can be checked by direct simulation or experiment.
  • Solvability appears linked to the existence of a time-independent low-dimensional effective description of the environment produced by the circuit itself.

Load-bearing premise

Representing the bath generated by the circuit evolution as a uniform MPS with finite bond dimension χ provides the correct and complete characterization of solvability.

What would settle it

A circuit whose dynamics satisfy an independent solvability criterion yet whose influence matrix cannot be represented by any finite-χ uniform MPS, or conversely an influence-solvable circuit whose subsystem evolution is not efficiently computable.

Figures

Figures reproduced from arXiv: 2606.12538 by Friedrich H\"ubner, Sun Woo P. Kim.

Figure 1
Figure 1. Figure 1: FIG. 1: We say a circuit is influence-solvable if its [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Using influence-solvability one can simplify [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Examples of scenarios applicable to influence-solvability. Any time-evolved observable [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Residual as function of [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Residual as function of [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Residual as function of [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Simulation of half-system setup Fig. [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Simulation of the two different partitioning setups Fig. [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
read the original 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 $\chi$. 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 $\chi=1$ 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 solvability conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces influence-solvable circuits in (1+1)D systems, defined via the influence matrix (representing the bath generated by circuit evolution) being a uniform MPS with finite bond dimension χ. It claims this class contains essentially all known solvable circuits (e.g., dual unitaries), derives necessary and sufficient local conditions from the fundamental theorem of MPS under open boundary conditions, and applies the framework to exhaustively classify brickwork circuits, finding new solvable examples for classical d≤3 and quantum d=2 that evade prior conditions.

Significance. If the central claims hold, the work supplies a unifying, MPS-based classification scheme for solvable circuits that enables efficient subsystem dynamics and systematic discovery beyond ad-hoc constructions. The explicit use of the MPS fundamental theorem to obtain local conditions, together with the concrete classification results for small d, would constitute a substantive advance in understanding tractable chaotic dynamics.

major comments (2)
  1. [derivation of local conditions] The weakest assumption—that a uniform finite-χ MPS representation of the influence matrix yields necessary and sufficient local conditions via the OBC fundamental theorem—requires explicit justification that non-uniform or infinite-χ representations cannot produce additional solvable circuits; without this, the necessity direction of the derived conditions remains open (abstract and § on derivation of local conditions).
  2. [introduction and comparison to prior work] The claim that influence-solvability 'essentially contains all known examples' is asserted but the manuscript provides no explicit mapping or exhaustive cross-check against the full literature list of dual-unitary generalizations; this is load-bearing for the 'systematic theory' framing (abstract).
minor comments (2)
  1. [definition of influence matrix] Notation for the influence matrix and its MPS tensors should be introduced with an explicit diagram or equation reference in the main text to aid readability.
  2. [classification results] The search procedure for the classification (enumeration method, symmetry reductions, verification that new circuits satisfy the local conditions) is only sketched; a supplementary section or appendix with pseudocode or explicit parameter counts would strengthen reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the scope and presentation of our results. We address each major comment below.

read point-by-point responses

  1. Referee: [derivation of local conditions] The weakest assumption—that a uniform finite-χ MPS representation of the influence matrix yields necessary and sufficient local conditions via the OBC fundamental theorem—requires explicit justification that non-uniform or infinite-χ representations cannot produce additional solvable circuits; without this, the necessity direction of the derived conditions remains open (abstract and § on derivation of local conditions).

    Authors: The derived local conditions are necessary and sufficient for the influence matrix to admit a uniform finite-χ MPS representation (under the open-boundary-conditions version of the fundamental theorem). By definition, influence-solvability is the class of circuits whose influence matrix possesses exactly this representation, which guarantees efficient subsystem dynamics. We agree that the manuscript would benefit from an explicit discussion of why we restrict attention to uniform finite-χ representations rather than allowing non-uniform or infinite-χ forms. In the revision we will add a clarifying paragraph in the section on the derivation of local conditions, explaining that non-uniform or infinite-χ representations fall outside the present definition because they would not in general permit the same efficient contraction or the same local algebraic conditions. revision: yes

  2. Referee: [introduction and comparison to prior work] The claim that influence-solvability 'essentially contains all known examples' is asserted but the manuscript provides no explicit mapping or exhaustive cross-check against the full literature list of dual-unitary generalizations; this is load-bearing for the 'systematic theory' framing (abstract).

    Authors: We accept that the current manuscript asserts the containment without supplying explicit mappings or a systematic cross-check against the literature on dual-unitary generalizations. In the revised version we will add a dedicated subsection (or an appendix table) that explicitly maps the principal known families—dual-unitary gates, their higher-dimensional and non-unitary generalizations, and the classical solvable circuits reported in the literature—to influence-solvable circuits, thereby making the claim concrete and supporting the “systematic theory” framing. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines influence-solvability directly as the property that the influence matrix is a uniform finite-χ MPS. Local conditions are obtained by applying the standard fundamental theorem of MPS (OBC version), an external result from tensor-network literature rather than a self-citation or fitted input. Exhaustive classification for small d proceeds by direct enumeration under these conditions. The claim that the class contains known solvable circuits is presented as externally verifiable against prior literature and does not reduce any prediction to the definition by construction. No step equates a derived quantity to its input via renaming, fitting, or self-referential closure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the newly introduced definition of influence-solvability and the applicability of the MPS fundamental theorem; no free parameters are indicated in the abstract.

axioms (1)
  • standard math Fundamental theorem of MPS for open boundary conditions yields necessary and sufficient local conditions
    Invoked to derive the local conditions for influence-solvability.
invented entities (2)
  • influence-solvable circuits no independent evidence
    purpose: Class of circuits whose influence matrix is a uniform MPS with finite χ
    New class defined to systematize solvability
  • influence matrix no independent evidence
    purpose: Represents the bath generated by the circuit's own evolution
    Central object in the new definition

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