Stabilizing Non-Abelian Topological Order against Heralded Noise via Local Lindbladian Dynamics

TL;DR

This work shows that fully local Lindbladian dynamics with heralded noise can stabilize steady-state phases exhibiting both Abelian and non-Abelian topological order in two dimensions. By designing measurement-and-feedback correction protocols that use heralded error information, the authors realize active dynamical phases where quantum information persists for times exponential in system size, separated from absorbing and partially active phases by first-order transitions. The Toric code serves as a warm-up to establish local correction techniques, while the central result demonstrates a non-Abelian $D_4$ steady-state TO stabilized by a generalized quasi-stabilizer framework and a non-Pauli stabilizer tableau, with two-way connectivity to the pure ground state. Imperfect heralding degrades topological order, but increasing heralding improves memory lifetimes, highlighting practical robustness for erasure-prone platforms. Overall, the paper provides a pathway to self-correcting quantum memories in 2D using strictly local dynamics under heralded noise, with implications for scalable quantum information protection.

Abstract

An important open question for the current generation of highly controllable quantum devices is understanding which phases can be realized as stable steady-states under local quantum dynamics. In this work, we show how robust steady-state phases with both Abelian and non-Abelian mixed-state topological order can be stabilized, in two spatial dimensions (2d), against generic ``heralded" noise using active dynamics that incorporate measurement and feedback, modeled as a \emph{fully local} Lindblad master equation. These topologically ordered steady states are two-way connected to pure topologically ordered ground states using local quantum channels, and preserve quantum information for a time that is exponentially large in the system size. Specifically, we present explicit constructions of families of local Lindbladians for both Abelian ($\mathbb{Z}_2$) and non-Abelian ($D_4$) topological order whose steady-states host mixed-state topological order when the noise is below a threshold strength. As the noise strength is increased, these models exhibit first-order transitions to intermediate mixed state phases where they encode robust classical memories, followed by (first-order) transitions to a trivial steady state at high noise rates. When the noise is imperfectly heralded, steady-state order disappears but our active dynamics significantly enhances the lifetime of the encoded logical information. To carry out the numerical simulations for the non-Abelian $D_4$ case, we introduce a generalized stabilizer tableau formalism that permits efficient simulation of the non-Abelian Lindbladian dynamics.

Figures (20)

• Figure 1: Steady-state phase diagrams for (a) the Toric code on a $48\times 48$ honeycomb lattice and (b) the non-Abelian $D_4$ model defined on three interpenetrating $48\times 48$ sized honeycomb lattices. The active dynamics removes the $X$ and $Z$ flags at rates $\gamma_x$ and $\gamma_z$, respectively. The heralded depolarizing noise rate is set to $\eta=1$. The background color indicates the combined density $\frac{1}{2}(n^f_X+n^f_Z)$ of flags evaluated at a late time ($t_f=100$ for the Toric code and $t_f=20$ for the $D_4$ model) at which the densities (for parameters away from the phase boundaries) have saturated to time-independent values (see Appendix \ref{['app:numerics']}). The logical error rates are computed for a select set of parameters depicted using the diamond-shaped symbols. These are colored according to $1-\frac{1}{2}(F_{|\widetilde{0}\rangle_L}+F_{|\widetilde{+}\rangle_L})$, which quantifies the steady-state logical error rate averaged over two distinct initial logical states. Each data point is obtained by averaging over $100$ independent Monte Carlo realizations. For both models, the active phase of the flags ($n^f_X,n^f_Z\approx 0$) leads to a topologically ordered steady-state phase, where both logical fidelities $F$ are close to $1$. This phase protects the initially encoded quantum information up to exponentially long times. In contrast, the partially active phases (only one type of flags has small density) can only protect classical information for long times, signaled by only one of the $F\approx 1$. The detailed time evolution for representative points in each of these phases, highlighted using cyan colored squares, is shown in Fig. \ref{['fig:TC-MC-dynamics']} and \ref{['fig:d4-densityVst']}.
• Figure 2: 2d Toric code on the honeycomb lattice: qubits live on the edges of a honeycomb lattice. A patch of $12\times 12$ size lattice is shown, with PBC in vertical and horizontal directions implemented according to Eq. \ref{['eq:tc-pbc']}. Edges involved in the vertex stabilizer $B_v$ and plaquette stabilizer $A_p$ are shown in red and green, respectively. The operators $l_H$ and $l_V$ run along non-contractible loops on the dual triangular lattice, defining the logical $\mathcal{Z}_H$ and $\mathcal{Z}_V$ operators, respectively.
• Figure 3: $X$ correction protocol for Toric code: The edges that have $X$ flags on them are shown using bold lines, and the vertex defects are depicted by black triangles. The initial configurations on the left side are updated to obtain the configuration on the right side. We illustrate processes in which stabilizer defects are present near the corrected vertex; the processes with no stabilizer defects act identically on the flags, but do not act on the qubits. (a) X-leaf move: Vertex $v$ has a single edge with an $X$ flag on it, and hosts a stabilizer defect. The defect is displaced by applying $X_j$ on the flagged edge $j$, after which the $X$ flag is removed. (b) X-loop move: (i) The two edges ($e_0$ and $e_1$) emanating from a vertex $v$ of the purple sublattice and bordering its northwest plaquette $p_{NW}$ have $X$ flags on them. The correction step in Eq. \ref{['eq:tc-x-loop']} removes these flags, raises new $X$ flags on the remaining edges of $p_{NW}$ (if they were not present to begin with; highlighted in green color), and displaces the vertex defect by applying $X_{e_0}$. (ii) The X-loop move can also be applied at vertex $\widetilde{v}$ which locally hosts a flag configuration $n^f_{X_{e_0}}=n^f_{X_{e_1}}=1$, and $n^f_{X_{e_2}}=0$ even though the flags do not form a closed loop. As a result of this operation, the vertex defects will be paired along the alternative path highlighted in green color. The loop moves are applied only at the vertices on the purple sublattice.
• Figure 4: $Z$ correction protocol for Toric code: The edges that have $Z$ flags are shown using bold wavy lines. The correction operation is applied to the blue plaquette in the top row to obtain the new configuration in the bottom row. Here, we illustrate processes in which stabilizer defects are moved by acting with $Z$ on appropriate edges (shown using dashed arrows); the processes with no stabilizer defects act identically on the flags, but do not apply the Pauli-$Z$ operator. (a) Z-leaf move: a plaquette defect occurs on a plaquette with a single flagged edge. The correction protocol applies $Z$ to this flagged edge, and subsequently erases the flag, moving the defect to a neighboring plaquette. (b) Z-loop moves: Four candidate processes corresponding to $Z$ flags on edges labeled by $(e_0,e_1)$,$(e_1,e_2)$,$(e_0,e_2)$, and $(e_0,e_1,e_2)$ are shown here. In each configuration, new $Z$ flags are added on edges $d_0$ and/or $d_1$ as described in the main text (refer to Eq. \ref{['eq:tc-z-loop']}).
• Figure 5: Time evolution of flag and defect densities, together with the logical fidelity, for the Toric code on a $48 \times 48$ honeycomb lattice for four representative points in the $\gamma_x$ - $\gamma_z$ phase diagram (see Fig. \ref{['fig:2dphase']}(a)). The probabilities of logical $X$ errors ($\equiv 1-F_{|\widetilde{0}\rangle_L}$) and logical $Z$ errors ($\equiv 1-F_{|\widetilde{+}\rangle_L}$) are computed by decoding the state at time $t$ using MWPM decoder (see Eqs. \ref{['eq:tc-logical-states']},\ref{['eq:tc-fidelity']}). The noise rate is set to $\eta=1$, with each data point representing an average over $500$ Monte Carlo runs. (a) Absorbing phase $(\gamma_x=4,\gamma_z=8)$: where flags rapidly reach their absorbing state $n^f_X=n^f_Z=1$, and all qubits become maximally mixed. The logical error rates $1-F_{|\widetilde{0}\rangle_L}$ and $1-F_{|\widetilde{+}\rangle_L}$ both saturate to their maximal value. (b) Partially X-active phase $(\gamma_x=15,\gamma_z=8)$: The density of $X$ flags and $B$ defects saturates at a small value, whereas $Z$ flags and $A$ defects proliferate, leading to small (large) values of logical $X (Z)$ error rate. (c) The analogous partially Z-active phase $(\gamma_x=4,\gamma_z=20)$ (d) Active phase $(\gamma_x=15,\gamma_z=20)$: here, both $X$ and $Z$ flag densities saturate to small values. Both logical $X$ (magenta colored curve) and logical $Z$ (cyan colored curve) error rates remain close to zero, and hence the quantum information can be recovered up to times exponentially large in system size.