Exact spectrum of the Floquet evolution operator of star-triangle integrable quantum circuits

Miao and Vernier (Quantum 7, 1160 (2023)) constructed the first integrable quantum circuits from the star-triangle relation rather than from Yang-Baxter vertex models: two-step Floquet evolutions $U_F(\phi)=U_1U_2$ on a periodic chain of $Q$-state qudits, commuting with two-parameter families of transfer matrices and possessing infinitely many local conserved charges, in two families — unitary $Q$-state Potts circuits and generally non-unitary $\mathbb{Z}_Q$ circuits from the Fateev–Zamolodchikov star-triangle solution. Unlike brick-wall circuits, however, $U_F(\phi)$ is not itself a transfer matrix of a solvable two-dimensional model, so its spectrum is not known. The problem, stated by the authors as 'the first natural question' left open by their work, is to compute the exact eigenvalue spectrum of $U_F(\phi)$ — e.g. via Bethe-ansatz-type functional equations through the Temperley–Lieb/six-vertex embedding for the Potts circuits and an adaptation of the Fateev–Zamolodchikov Bethe equations for the $\mathbb{Z}_Q$ circuits — together with a completeness analysis. The problem has remained open since the source's publication.

Background

A quantum circuit is a discrete-time model of many-body dynamics: a Floquet evolution operator $U_F=U_1U_2$ applied stroboscopically, where $U_1$ is a product of one-site gates and $U_2$ a product of mutually commuting two-site gates on a periodic chain of $L$ qudits ($Q$-level spins, Hilbert space $(\mathbb{C}^Q)^{\otimes L}$, with clock operators $X,Z$ obeying $XZ=\omega ZX$, $\omega=e^{2\pi i/Q}$). A circuit is called integrable if it admits an extensive family of independent local conserved charges; this strongly constrains its thermalization and transport, exactly as for integrable Hamiltonians.

Integrable circuits were previously constructed as brick-wall Trotterizations (brick-patterned discrete-time Suzuki–Trotter decompositions of continuous-time evolution) of Yang–Baxter integrable spin chains. There the Floquet operator is essentially the row-to-row transfer matrix — the operator whose eigenvalues encode the free energy of an associated two-dimensional vertex model — of a model satisfying the Yang–Baxter equation, the consistency condition that makes transfer matrices commute, so its spectrum follows from the Bethe ansatz, the standard scheme for diagonalizing such transfer matrices through functional equations for rapidities ('Bethe roots').

Miao and Vernier instead built integrable circuits from the other great solvability principle of two-dimensional statistical mechanics, the star-triangle relation (STR). The STR is a local linear identity between Boltzmann weights of edge-interaction ('spin') models on two dual lattices; it plays for edge models the role the Yang–Baxter equation plays for vertex models and underlies the exact solvability of, for instance, the Ising and chiral Potts models (the $\mathbb{Z}_Q$-symmetric edge models whose Bethe-ansatz solution lives on a high-genus algebraic curve). From STR solutions they constructed two-parameter families of mutually commuting transfer matrices $T(\lambda,\mu;\phi)$ which, at special parameter values, generate the gates of circuits whose Floquet operator $U_F(\phi)$ commutes with the entire family; expanding $T$ around those points yields infinitely many local conserved charges. Two families result: (a) $Q$-state Potts circuits, with symmetric-group $S_Q$ symmetry, unitary at the admissible gate parameters $\phi$, whose charges are expressed through the Temperley–Lieb algebra (the diagrammatic algebra generated by operators $e_j$ with $e_j e_{j\pm1} e_j=e_j$ and $e_j^2=\delta e_j$, which underlies the six-vertex model — the archetypal exactly solved vertex model — and the Potts structures); this proved the integrability conjecture of Lotkov et al. for the kicked Potts circuit; and (b) $\mathbb{Z}_Q$ circuits from the Fateev–Zamolodchikov self-dual STR solution (the $\mathbb{Z}_N$-symmetric spin model whose Hamiltonian limit is the spin-1 chain related to the 19-vertex model, a solvable vertex model generalizing the six-vertex model to spin-1 variables), generally non-unitary and, for $Q=4$, related to the critical Ashkin–Teller chain (two coupled Ising models with a four-spin interaction).

What the construction does not deliver is the spectrum. Unlike the brick-wall case, $U_F(\phi)$ is not itself a member of the commuting transfer-matrix family — the 'staggering' producing the circuit geometry acts on the internal parameters of the individual R-matrices (the local operators out of which the transfer matrices are built) rather than on site-alternating spectral parameters (the per-site inhomogeneities $\lambda,\mu$ labelling the transfer-matrix family) — so the known Bethe equations of the underlying two-dimensional models do not directly diagonalize it. In their conclusion the authors state the exact computation of the spectrum of $U_F(\phi)$ as 'the first natural question', with suggested routes: for the Potts circuits, a Temperley–Lieb/six-vertex Bethe ansatz (compare the Bethe equations derived for the three-state integrable Potts chain) and, for the $\mathbb{Z}_Q$ circuits, an adaptation of the Bethe-ansatz-type equations known for Fateev–Zamolodchikov-type $\mathbb{Z}_Q$-invariant models. The same conclusion lists four further open directions — quench dynamics and late-time steady states (integrable quenches via the boundary star-triangle relation, in the framework of the generalized Gibbs ensemble — the maximum-entropy steady state fixed by the complete set of conserved charges — for discrete-time evolution), the physical status of the non-unitary circuits (whether they define completely positive trace-preserving maps, the Markovian quantum channels describing open-system dynamics, and are thus relevant to open quantum systems), circuits from more general STR solutions, and continuum field-theory limits — which are sibling research directions rather than part of the spectral question above.

Problem Statement

For the two families of integrable quantum circuits constructed from the star-triangle relation by Miao and Vernier (Quantum 7, 1160 (2023)) — (a) the $Q$-state Potts circuits, unitary at the admissible values of the gate parameter $\phi$, and (b) the generally non-unitary $\mathbb{Z}_Q$ circuits obtained from the Fateev–Zamolodchikov star-triangle solution — compute the exact eigenvalue spectrum of the Floquet evolution operator $U_F(\phi)=U_1(\phi)U_2(\phi)$ on a periodic chain of $L$ qudits, for the genuinely interacting cases $Q\ge 3$, at generic $\phi$ and — for the Potts circuits, where unitarity holds only at special admissible values — also at those unitary points.

Concretely, a solution must:

1. derive Bethe-ansatz-type functional equations (or an equivalently rigorous exact characterization) satisfied by the eigenvalues of $U_F(\phi)$ at finite $L$, for each circuit family; and

2. establish completeness: show that the solutions of these equations enumerate the full spectrum of $U_F(\phi)$, with any exceptional or degenerate sectors precisely characterized.

The routes anticipated in the source — the Temperley–Lieb/six-vertex Bethe-ansatz embedding for the Potts circuits and an adaptation of the Fateev–Zamolodchikov Bethe equations for the $\mathbb{Z}_Q$ circuits — are suggested methods, not restrictions on the admissible answer; any exact method of equal rigor is acceptable. A complete resolution covers both families; an exact solution for either family alone (for instance only the Potts circuits, or a single value such as $Q=3$) is well-defined partial progress. The $Q=2$ cases, where the Potts model reduces to the Ising model and the circuit dynamics is of the kicked-Ising type solvable by free-fermion (Jordan–Wigner) techniques, are not part of the target.

References

1. Yuan Miao and Eric Vernier. Integrable Quantum Circuits from the Star-Triangle Relation. Quantum 7, 1160 (2023). DOI: 10.22331/q-2023-11-03-1160. arXiv:2302.12675. https://doi.org/10.22331/q-2023-11-03-1160
2. A. I. Lotkov, V. Gritsev, A. K. Fedorov, and D. V. Kurlov. Floquet integrability and long-range entanglement generation in the one-dimensional quantum Potts model. Phys. Rev. B 105, 144306 (2022). DOI: 10.1103/PhysRevB.105.144306. arXiv:2110.09559. https://doi.org/10.1103/PhysRevB.105.144306
3. Giuseppe Albertini, Srinandan Dasmahapatra, and Barry M. McCoy. Spectrum and Completeness of the Integrable 3-State Potts Model: a Finite Size Study. Int. J. Mod. Phys. A 7, 1-53 (1992). DOI: 10.1142/S0217751X92003719. https://doi.org/10.1142/S0217751X92003719
4. Giuseppe Albertini. Bethe-ansatz type equations for the Fateev-Zamolodchikov spin model. J. Phys. A: Math. Gen. 25, 1799-1813 (1992). DOI: 10.1088/0305-4470/25/7/021. https://doi.org/10.1088/0305-4470/25/7/021
5. V. A. Fateev and A. B. Zamolodchikov. Self-dual solutions of the star-triangle relations in ZN-models. Phys. Lett. A 92, 37-39 (1982). DOI: 10.1016/0375-9601(82)90736-8. https://doi.org/10.1016/0375-9601(82)90736-8
6. Yu-Kui Zhou. Fateev-Zamolodchikov and Kashiwara-Miwa models: boundary star-triangle relations and surface critical properties. Nucl. Phys. B 487, 779-794 (1997). DOI: 10.1016/S0550-3213(96)00674-8. https://doi.org/10.1016/S0550-3213(96)00674-8
7. Lucas Sá, Pedro Ribeiro, and Tomaž Prosen. Integrable nonunitary open quantum circuits. Phys. Rev. B 103, 115132 (2021). DOI: 10.1103/PhysRevB.103.115132. https://doi.org/10.1103/PhysRevB.103.115132
8. Eric Vernier, Hsiu-Chung Yeh, Lorenzo Piroli, and Aditi Mitra. Strong Zero Modes in Integrable Quantum Circuits. Phys. Rev. Lett. 133, 050606 (2024). DOI: 10.1103/PhysRevLett.133.050606. arXiv:2401.12305. https://doi.org/10.1103/PhysRevLett.133.050606
9. Friedrich Hübner, Eric Vernier, and Lorenzo Piroli. Generalized hydrodynamics of integrable quantum circuits. SciPost Phys. 18, 135 (2025). DOI: 10.21468/SciPostPhys.18.4.135. arXiv:2408.00474. https://doi.org/10.21468/SciPostPhys.18.4.135
10. Dávid Szász-Schagrin, Daniele Cristani, Lorenzo Piroli, and Eric Vernier. Construction and simulability of quantum circuits with free fermions in disguise. Quantum Sci. Technol. 11, 015044 (2026). DOI: 10.1088/2058-9565/ae390d. arXiv:2509.22585. https://doi.org/10.1088/2058-9565/ae390d
11. Zhao Zhang. The Yang-Baxter Equation for the Chiral Potts Model and Integrable Parafermions (2026). arXiv:2606.11146. https://arxiv.org/abs/2606.11146
12. Yunfeng Jiang, Chang Liu, Yu Wu, and Yang Zhang. Gate Parameter Lee-Yang Zeros and Dynamical Phases in Quantum Circuits (2026). arXiv:2605.29838. https://arxiv.org/abs/2605.29838
13. Bruno Bertini, Pieter W. Claeys, and Tomaž Prosen. Exactly solvable quantum many-body dynamics from space-time duality. Rev. Mod. Phys. 98, 025001 (2026). DOI: 10.1103/yx73-dk86. arXiv:2505.11489. https://doi.org/10.1103/yx73-dk86