Classification of integrable interaction-round-a-face models of qubits and of their integrable unitary boundary processes

The deformed rule-54 reversible cellular automaton (RCA54) of Paletta and Prosen (2026) is an interaction-round-a-face (IRF, or face-type) brickwork quantum circuit of qubits whose integrability is anchored in a transfer matrix commuting with the discrete-time propagator; the proofs, however, are perturbative in the deformation parameters or exact only at fixed parameter values. The authors leave open the systematic face-type theory: (i) a complete classification of integrable IRF models of qubits ($d=2$), including whether their four-parameter deformation is the most general integrable deformation of RCA54; (ii) an IRF analog of the Sklyanin reflection algebra adapted to unitary discrete-time dynamics; and (iii) the classification of integrable unitary boundary processes for such open circuits.

Background

A reversible cellular automaton (RCA) is a lattice dynamical system whose local update rule is invertible; rule 54 (RCA54), introduced by Bobenko, Bordemann, Gunn and Pinkall in 1993, is a two-state deterministic RCA on a one-dimensional spatial lattice with discrete time, viewable as a minimal soliton theory in $1+1$ dimensions, and has become a workhorse of exactly solvable nonequilibrium statistical mechanics (Buča, Klobas and Prosen, 2021). When formulated as a two-dimensional lattice model, RCA54 is naturally an interaction-round-a-face (IRF, or face-type) model: Boltzmann weights are attached to elementary faces (plaquettes) and indexed by the corner variables (the spin values $0,1$ sitting on the lattice sites), rather than to vertices as in the familiar six-vertex model (Prosen, 2021). This is the structural reason RCA54 does not fit the standard vertex-based quantum inverse scattering method.

Integrability of a discrete-time many-body dynamics means that the propagator $\mathbb{U}$ commutes with an infinite family of mutually commuting conserved charges. In the vertex world such structures are built from a Lax operator (a local spectral-parameter-dependent operator whose ordered product along the chain defines the transfer matrix) and a transfer matrix whose commutativity follows from the Yang–Baxter equation (a consistency condition on local $R$-matrices ensuring commuting transfer matrices), and open chains are handled by Sklyanin's reflection algebra (1988): boundary $K$-matrices solving reflection equations generate commuting double-row transfer matrices and hence integrable boundary conditions. For two-state vertex systems this side is essentially closed: the classification of $4\times 4$ (qubit) Yang–Baxter operators goes back to Hietarinta (see the modern algebraic treatment by Maity, Singh, Padmanabhan and Korepin, 2024), de Leeuw, Paletta, Pribytok, Retore and Ryan (2020) classified two-qubit nearest-neighbour integrable interactions, and García Fernández, Paletta and Retore (2026) classified integrable open quantum circuits of arbitrary geometry for six- and eight-vertex $R$-matrices. No face-type counterpart of any of this exists.

For RCA54 itself, an integrability-preserving deformation was constructed and a quantum Lax operator built in earlier work (Gombor and Pozsgay, 2024, building on the medium-range (few sites beyond nearest neighbours) integrability framework of Gombor and Pozsgay, 2021), but the dynamical map itself could not be recovered from that Lax operator or its transfer matrix. Paletta and Prosen (2026) closed this gap for a quantum/stochastic deformation of RCA54 parametrized, under periodic boundary conditions, by four complex parameters $\alpha,\beta,\gamma,\delta$: they showed the shortest nontrivial conserved charge $Q_6$ has range six, embedded it into an infinite commuting hierarchy generated by logarithmic derivatives of a transfer matrix, proved $[t(u),\mathbb{U}]=0$ via an additional intertwining operator, and constructed the exact solution of the non-equilibrium steady state (NESS, the fixed point of the boundary-driven stochastic dynamics) by a staggered patch matrix product ansatz (a representation of the steady state as a product of matrices associated with local space-time patches). Crucially for the present problem, these constructions are proven only perturbatively in the deformation parameters (charges $Q_6,Q_{10},Q_{14}$, Lax operator to order $u^4$) or nonperturbatively at fixed numerical values of the parameters, while integrability is expected to hold for arbitrary complex parameters.

Two adjacent literatures delimit the open question. First, reflection equations for equilibrium face models do exist: Behrend, Pearce and O'Brien (1995), Behrend and Pearce (1995, 1996) and Zhou and Batchelor (1995) constructed commuting double-row transfer matrices for IRF models with fixed or free boundary conditions and solved the boundary Yang–Baxter equation for Andrews–Baxter–Forrester (ABF) and ADE face models. These concern passive boundary conditions of two-dimensional equilibrium face models whose face variables live on ADE graphs, not dynamical unitary gates for discrete-time qubit circuits, and they contain no classification of two-state face dynamics. Second, other exactly solvable quantum cellular automata of face type are known in isolation — e.g. the free-fermion-related Goldilocks rules (Hillberry, Piroli, Vernier, Yunger Halpern, Prosen and Carr, 2026) and the three-state IRF automata of Klobas and Prosen (2022) whose conservation laws were recently analyzed by Prosen (2026) — but none of these works attempts a classification. The deformed RCA54 is thus a single, tantalizing data point in an unclassified landscape.

Problem Statement

Establish the bulk-and-boundary integrability classification of two-state interaction-round-a-face (IRF) dynamics, in the setting exemplified by the deformed rule-54 reversible cellular automaton. Concretely, deliver: (1) Bulk classification. A complete classification, up to gauge equivalence of face weights (equivalence under site-dependent redefinitions of the corner variables), of integrable IRF models of qubits — face models whose corner variables take the two values $0,1$, whose local face weights satisfy the face (IRF) Yang–Baxter/star–triangle relations ensuring commuting transfer matrices, including the discrete-time dynamical realizations as brickwork circuits of face gates (odd and even bonds updated in alternating sub-steps) for which the transfer matrix commutes with the propagator. This includes deciding whether the four-parameter ($\alpha,\beta,\gamma,\delta$) deformation of RCA54 constructed by Paletta and Prosen is the most general integrable deformation of the rule-54 update, and placing all known isolated solutions — the deformed RCA54 family, free-fermion/Goldilocks-type rules, and any Ising-type or hard-square face limits — within the classification. (2) Boundary theory. A formulation of the IRF analog of the Sklyanin reflection algebra adapted to unitary discrete-time face dynamics: algebraic relations on boundary face weights (boundary $K$-data) that guarantee commutation of the corresponding double-row-type transfer matrices and conservation of the charges for open chains; and, within this framework, a classification of the integrable unitary boundary processes for two-state IRF circuits, i.e. the boundary gates preserving integrability. Deliverables (1) and (2) jointly constitute one objective — the complete inventory of solvable two-state face dynamics with and without boundaries — with (2) built on the bulk framework fixed by (1); both deliverables are required.

References

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