Self-correction phase transition in the dissipative toric code

TL;DR

Problem: to realize long-lived quantum information in an open system using a local, time-continuous decoder. Approach: model a dissipative toric code via a Lindblad master equation coupled to a 2D CA field that steers anyon motion, and map steady-state behavior to a topological-order criterion and a phase diagram in the rate parameters $\gamma_1$, $\gamma_2$, $\gamma_3$. Findings: there exists a self-correcting steady-state phase below a critical ratio $\gamma_1^c/\gamma_2 \approx 0.008$ (for typical $\gamma_3$), with a 2D CA field sufficing; an operational definition of topological order via circuit-depth captures the phase transition; memory robustness shows an optimal CA-update rate. Significance: demonstrates a practical, scalable path to passive quantum memories in open quantum systems, with locality enabling integration into qubit-control hardware via $O(1)$ CA updates.

Abstract

We analyze a time-continuous version of a cellular automaton decoder for the toric code in the form of a Lindblad master equation. In this setting, a self-correcting quantum memory becomes a thermodynamical phase of the steady state, which manifests itself through the steady state being topologically ordered. We compute the steady state phase diagram, finding a competition between the error correction rate and the update rate for the classical field of the cellular automaton. Strikingly, we find that self-correction of errors is possible even in situations where conventional quantum error correction does not have a finite threshold.

Figures (5)

• Figure 1: Schematic of the toric code. Left: Spins are placed on the surface of a torus, with the highlighted nontrivial loop corresponding to an operation changing the logical value of the stored information. Right: The surface is structured according to a checkerboard pattern of vertex operators $A_v$ and plaquette operators $B_p$ involving the adjacent four spins of each vertex and plaquette, respectively.
• Figure 2: Logical error probability $p_\varepsilon$ in the steady state depending on the error rate $\gamma_1$ for different system sizes $L\times L$. The field-update rate is taken as $\gamma_3 = 10\,\gamma_2$. Each data point is an average obtained from $10^{3}-$simulation runs. The simulation points to a critical error rate of about $\gamma_{1}=10^{-2}\,\gamma_2$, below which logical errors are increasingly suppressed with system size.
• Figure 3: Mean anyon density $n$ for various system sizes $L\times L$ as a function of the error rate $\gamma_{1}$. The inset depicts the variance of the anyon density $\sigma^2_n$. The variance decreases with increasing system size.
• Figure 4: (a) Mean circuit depth $d$ normalized to the system size $L\times L$ as a function of the error rate $\gamma_1$ with fixed $\gamma_3 = 10\,\gamma_2$. Above the self-correction transition, the normalized circuit depth shows a modified scaling with system size. (b) Variance of the circuit depth $\sigma_d^2$ normalized to the system size. Below the transition, the normalized variance vanishes in the thermodynamic limit, while it reaches a finite value after the transition.
• Figure 5: Phase diagram depicting the critical error rate $\gamma_1^c$ depending on the CA field-update rate $\gamma_3$. The shaded region indicates the self-correcting phase in which errors can successfully be corrected. This suggests an optimal rate of CA field-update beyond which the critical error rate starts declining.