Spin stiffness and resilience phase transition in a noisy toric-rotor code

Abstract

We use a quantum formalism for the partition function of the classical $XY$ model to identify a resilience phase transition in a noisy toric-rotor code. Specifically, we consider the toric-rotor code under phase-shift noise described by a von Mises probability distribution and show that the fidelity between the final state after noise and the initial state is proportional to the partition function of the $XY$ model. We map the temperature of the $XY$ model to the width of the noise in the toric-rotor code, such that a Kosterlitz--Thouless phase transition at a critical temperature $T_{c}$ corresponds to a mixed-state phase transition at a critical width $σ_c$. To characterize this phase transition, we develop a quantum formalism for the spin stiffness in the $XY$ model and show that it is mapped to the gate fidelity in the logical subspace of the toric-rotor code. In particular, we introduce a topological order parameter that characterizes the resilience of the toric-rotor code to decoherence within the logical subspace. We show that the logical subspace does not exhibit complete resilience to noise, which is a necessary condition for correctability. However, it exhibits partial resilience to noise for widths less than $σ_c\approx 0.89$, where the resilience order parameter takes values near $1$ and then drops to zero at $σ_c$. We also use our results to shed light on the correctability of toric-rotor codes in higher dimensions $d > 2$. Our work shows that the quantum formalism for partition functions provides a mathematically rigorous framework for studying correctability in continuous-variable quantum codes.

Figures (7)

• Figure 1: Two-dimensional square lattice of the toric-rotor code. Quantum rotors (green circles) reside on the edges of the lattice, with a fixed orientation assigned to each edge. A blue square represents a face stabilizer, constructed as the product of the corresponding operators around the face; for example, $B_f(m)=X_1(m)X_2(-m)X_3(-m)X_4(m)$. A red square represents a vertex stabilizer, constructed as the product of the corresponding operators on the edges incident to the vertex; for example, $A_v(\phi)=Z_1(\phi)Z_2(\phi)Z_3(-\phi)Z_4(-\phi)$.
• Figure 2: Non-local operators in the toric-rotor code. The red lines denote the non-contractible loops $T_x(m)$ and $T_y(m')$ along the horizontal and vertical lattice directions, formed by products of the corresponding operators. The blue lines denote the operators $W_{\bar{x}}(\phi)$ and $W_{\bar{y}}(\phi')$, defined along horizontal and vertical paths of the dual lattice and intersecting the non-contractible loops.
• Figure 3: Square-lattice representation of the two-dimensional $XY$ model. An orientation is assigned to each lattice edge, and the partition function is rewritten by replacing vertex variables $\theta$ with edge variables $\Theta_e$. Each edge variable $\Theta_e$ (blue squares) is associated with an oriented edge $e$ connecting vertices $i$ and $j$.
• Figure 4: Geometric representation of the constraints on the edge variables in the rewritten partition function [Eq. (\ref{['partitionfunction']})]. The symbol $\overset{f}{\delta_{2\pi}}$ (diamond-shaped region) denotes the local plaquette constraint, enforcing that the oriented sum of edge variables around any closed loop equals an integer multiple of $2\pi$. The horizontal and vertical colored bands indicate nonlocal constraints imposed by periodic boundary conditions.
• Figure 5: Schematic illustration of twisted periodic boundary conditions for the $XY$ model and its interpretation in the toric-rotor code. The twist is introduced by imposing a phase shift $\phi$ on the edges crossing a specified boundary (indicated by the red shaded bar corresponding to a non-contractible loop $C_{\bar{y}}$). For these edges, which $\delta_e = 1$, the interaction term in the Hamiltonian changes to $-J \cos(\Theta_e - \phi)$ where $\Theta_e =\theta_i -\theta_j$. Using the quantum formalism, the twist corresponds to applying $W_{\bar{y}}(\phi)$ to the initial state $|\Psi_{0,0}\rangle$ in the presence of noise.