Skeleton of isometric Tensor Network States for Abelian String-Net Models

TL;DR

The work constructs skeletons of isometric tensor-network states (isoTNS) for abelian string-net models, enabling efficient quantum preparation and tractable classical evaluation of many observables through mapping to stochastic automata. By enforcing virtual symmetries and isometry, the authors connect fixed-point topological orders via analytically tractable, continuous deformations, revealing phase transitions beyond anyon condensation. They demonstrate explicit isoTNS paths linking $Z_N$ toric codes, double-semion, and symmetry-enriched orders, and show how generalized Pauli strings up to arbitrary weight can be computed classically in many cases. The framework provides a versatile, testable platform for benchmarking quantum processors and exploring a rich landscape of topological and SET phases with finite correlation length. It also highlights how extending to higher multipole-conserving processes yields new critical behaviors and deeper connections between stochastic dynamics and topological order.

Abstract

We construct parametrized isometric tensor network states -- referred to as skeletons -- that allow us to explore phases of abelian topological order and can be efficiently implemented on quantum processors. We obtain stable finite correlation length deformations of string-net fixed points, which are constructed both by conserving virtual symmetries of the tensor and by imposing local isometry constraints. They connect distinct topological phases via a shared critical point, thereby providing analytically tractable examples of phase transitions beyond anyon condensation. By mapping such classes of 2D tensor networks to 1D stochastic automata with local update rules, we show that expectation values of generalized Pauli strings of arbitrary weight can be efficiently computed using classical methods. Therefore these skeletons not only serve as an organizing principle for abelian topological order but also provide a non-trivial testbed for quantum processors.

Figures (4)

• Figure 1: Skeleton of isometric Tensor Network States. Starting from abelian string-net models, we construct paths of isometric tensor networks within the topological phase of matter, connected by an imaginary time evolution. At some end points (red dots), a critical state is reached from different topologically ordered phases, realizing a quantum phase transition. For local Hilbert space dimension $N = 4$ possible phases include pairs of toric code ($\mathbb{Z}_2$), pairs of double-semion (DS) states, or the $\mathbb{Z}_4$ toric code. The subscript $f$ denotes phases with non-trivial symmetry fractionalization under an anti-unitary symmetry. The dashed path is further investigated in Fig.\ref{['fig:Path']}. The isometric tensor network is illustrated on the right.
• Figure 2: Mapping to stochastic automata. The normalization property of the single-line $W$-matrix (left) allows us to identify it with the two-local update rule of a stochastic automaton (right). From this, a tensor network state with an open boundary can be interpreted as a superposition of all paths of this stochastic process, where one spatial direction of the quantum state becomes the time direction of the automaton. Diagonal operators such as $Z_j$ on site $j$ can be evaluated with classical Monte Carlo methods from the local state $n_j$ in the stochastic circuit. Generalized Pauli string operators such as loop operators $(XX^\dagger)^{\otimes2}$ (red) can be mapped to functions of the classical state at different points in time and space $f(n_{\{j\}})$, and thereby efficiently evaluated as well.
• Figure 3: Path crossing multiple string-net phases. We track the correlation length $\xi$ along the path shown in Fig.\ref{['fig:Skeletons']}; the tensor network passes from a pair of decoupled double-semion models (yellow) to a pair of $\mathbb{Z}_2$ toric codes (violet), which are subsequently coupled to a $\mathbb{Z}_4$ toric code (purple). If we further conserve a global anti-unitary $\mathbb{Z}_2^T$ symmetry, this phase is distinguished from another phase with the same topological order but non-trivial symmetry fractionalization on the anyons labeled $\mathbb{Z}_{4,f}$ (turquoise). The correlation length is zero at the fixed points and diverges at the critical points, where a conservation law emerges in the associated stochastic circuit, leading to algebraic decay of correlations in one direction. We also show a membrane order parameter $\langle O_{e_4} \rangle$ which distinguishes $\mathbb{Z}_4$ and $\mathbb{Z}_{4,f}$.
• Figure S1: Dipole isoTNS. (Left column) By splitting up a $9$-level edge degree of freedom into two qutrits, the corresponding normalized $W$-matrix can be mapped to a 4-local gate of a stochastic process. The weights can be chosen in such a way as to conserve a global $U(1)$ charge $Q = \sum_j q_j$ in the time evolution (red) or the charge $Q$ alongside the associated dipole moment $P = \sum_j jq_j$ (blue). (Right column) These conservation laws lead to diffusive equal-position correlations $\sim r^{-1/2}$ for a conserved charge and subdiffusive correlations $\sim r^{-1/4}$ for conserved charge and dipole moment which are reflected in the correlations of the respective isoTNS. The numerical results are evaluated starting from an open boundary of two edges meeting at a corner (lower left inset). The charge-conserving critical reference state given by $W^Q$ has isoTNS deformations of its own consisting of paths which respect charge conservation of the stochastic process; it can be continuously connected to a topological phase such as the $\mathbb{Z}_9$ toric code or to the dipole critical state by passing through a path on the $N = 9$ isoTNS skeleton (top right inset).