Exact spectrum of the Floquet evolution operator of star-triangle-derived integrable quantum circuits (Potts and $\mathbb{Z}_Q$ circuits)

Miao and Vernier [Quantum 7, 1160 (2023)] constructed integrable discrete-time quantum circuits from the star-triangle relation of two-dimensional exactly solvable lattice models rather than from Yang-Baxter vertex models: $Q$-state Potts circuits on chains of qudits (proving an integrability conjecture of Lotkov et al.) and novel $\mathbb{Z}_Q$ circuits from the Fateev-Zamolodchikov self-dual star-triangle solution. Integrability is certified by mutually commuting two-parameter transfer matrices, but the paper poses as its first open question whether the spectrum of the Floquet evolution operator $\mathbf{U}_{\mathrm{F}}(\phi)$ can be computed exactly using the toolbox of integrability. The problem is to derive exact functional or Bethe-ansatz-type equations for these circuits --- exploiting the Temperley-Lieb embedding of the Potts circuits or adapting the Fateev-Zamolodchikov Bethe equations, or any other exact method --- yielding all eigenvalues of $\mathbf{U}_{\mathrm{F}}(\phi)$ for finite chains, together with a completeness analysis. The problem remains unresolved in the follow-up literature; the other directions raised in the same conclusion --- integrable quenches, the completely-positive-trace-preserving (CPTP) status of the non-unitary circuits, more general star-triangle solutions, and continuum limits --- are distinct problems and lie outside the scope of this task.

Background

A discrete-time quantum circuit is a unitary evolution operator built as a product of local gates; when the gates repeat periodically it is the stroboscopic (Floquet) evolution operator of a time-periodic Hamiltonian, $\mathbf{U}_{\mathrm{F}} = \mathcal{T}e^{-i\int_0^{2\tau}H(t)dt}$. For two-step protocols with on-site and nearest-neighbour pieces this becomes $\mathbf{U}_{\mathrm{F}} = U_1U_2$ with $U_1=\prod_m v_m$ and $U_2=\prod_m w_{m,m+1}$. A circuit is called integrable if it possesses an extensive family of local conserved charges commuting with $\mathbf{U}_{\mathrm{F}}$. Previously known integrable circuits are brickwork arrangements obtained by Trotterization (time discretization) of an integrable spin chain, with gates built from $R$-matrices solving the Yang-Baxter equation of vertex models; for such circuits exact spectra are available in several cases (e.g. the Trotterized XXZ chain).\n\nThe star-triangle relation (STR) is the integrability mechanism of a complementary class of two-dimensional exactly solvable classical lattice models (Ising, chiral Potts, Fateev-Zamolodchikov $\mathbb{Z}_N$ models). It states that the sum over an internal spin of a product of three edge weights $K(\theta;i,m)$ equals a prefactor times a product of three dual weights $K(\pi-\theta;j,k)$ arranged as a triangle; it guarantees commutation of the row transfer matrices, i.e. the exact solvability of the model. The Fateev-Zamolodchikov solution [Phys. Lett. A 92, 37 (1982)] is a self-dual $\mathbb{Z}_N$-symmetric STR solution.\n\nMiao and Vernier built, for the first time from the STR rather than from vertex models, integrable circuits on a chain of $L$ qudits ($Q$-level spins, Hilbert space $(\mathbb{C}^Q)^{\otimes L}$, periodic boundary conditions). Writing $X,Z$ for the clock operators with $XZ=\omega ZX$, $\omega=e^{2\pi i/Q}$, the circuits take the $\mathbb{Z}_Q$-symmetric form $U_1=\prod_m\sum_{a}u_a X_m^a$ and $U_2=\prod_m\sum_a v_a (Z_m^\dagger Z_{m+1})^a$. The construction provides a two-parameter family of inhomogeneous 'two-row' transfer matrices $t(u,\phi)$ of the STR-solved spin model which, at special values of the spectral parameters, collapse onto the circuit itself (up to a diagonal gauge factor), so that $[t(u,\phi),\mathbf{U}_{\mathrm{F}}(\phi)]=0$ and logarithmic derivatives of $t$ yield local conserved charges. Two families were worked out: (a) $Q$-state Potts circuits, unitary for suitable values of the Floquet angle $\phi$; their integrability had been conjectured by Lotkov, Gritsev, Fedorov and Kurlov on the basis of the Temperley-Lieb algebra (the algebra generated by $e_m$ with $e_m^2\propto e_m$ and braid-type relations, underlying Potts-type models), and was proven from the Potts STR; for $Q=3$ they connect to the Zamolodchikov-Fateev 19-vertex model at the root of unity $q=e^{i\pi/3}$; (b) $\mathbb{Z}_Q$ circuits from the Fateev-Zamolodchikov self-dual STR solution, which are non-unitary for generic $\phi$ and for $Q=4$ relate to the critical Ashkin-Teller chain (a two coupled-Ising-chain model).\n\nWhat is missing is the spectral side. The commuting objects are staggered, inhomogeneous two-row transfer matrices, for which no Baxter-type functional relations (a closed system of $TQ$-type equations, fusion and inversion identities) are known, and the eigenvalues of $\mathbf{U}_{\mathrm{F}}(\phi)$ have not been extracted from the two-parameter family. Bethe-ansatz-type equations are known for the Hamiltonian limits and related row-transfer-matrix problems --- e.g. for the three-state Potts model and for the Fateev-Zamolodchikov spin model (Albertini, J. Phys. A 25, 1799 (1992)) --- and for brickwork vertex-model circuits, but none of these directly diagonalizes $\mathbf{U}_{\mathrm{F}}(\phi)$ of the STR-derived circuits. The source paper therefore poses, as its first open question, whether this spectrum can be computed exactly with the toolbox of integrability.

Problem Statement

Compute the exact spectrum of the Floquet evolution operator $\mathbf{U}_{\mathrm{F}}(\phi)=U_1(\phi)U_2(\phi)$ of the star-triangle-derived integrable quantum circuits constructed by Miao and Vernier, on a chain of $L$ qudits with Hilbert space $(\mathbb{C}^Q)^{\otimes L}$ and periodic boundary conditions, for arbitrary finite $L$, $Q$, and Floquet angle $\phi$. The two constructed families are both in scope: the $Q$-state Potts circuits (unitary at the admissible $\phi$, with $U_1=\prod_m\sum_a u_a X_m^a$, $U_2=\prod_m\sum_a v_a(Z_m^\dagger Z_{m+1})^a$ and the Lotkov et al. parameter set), and the $\mathbb{Z}_Q$ circuits from the Fateev-Zamolodchikov self-dual star-triangle solution (non-unitary for generic $\phi$).\n\nA complete answer consists of: (1) closed functional relations or Bethe-ansatz-type equations --- for instance for the mutually commuting two-parameter transfer matrices $t(u,\phi)$ whose special points reproduce $\mathbf{U}_{\mathrm{F}}(\phi)$, exploiting the Temperley-Lieb embedding for the Potts circuits or adapting the known Fateev-Zamolodchikov Bethe equations for the $\mathbb{Z}_Q$ circuits, or any other exact method --- from which the eigenvalues of $\mathbf{U}_{\mathrm{F}}(\phi)$ are obtained exactly at finite size; and (2) a completeness analysis showing that the solutions of these equations account for all $Q^L$ eigenvalues up to exact symmetry degeneracies, including in the non-unitary $\mathbb{Z}_Q$ case. The question of whether such an exact computation is possible at all using the toolbox of integrability is itself part of the problem. The two families may be treated separately; a solution for either family, or for precisely delimited parameter regimes, is a valid partial answer whose scope must be stated explicitly.

References

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