Fast offline decoding with local message-passing automata

TL;DR

This work presents a fast, fully local offline decoder for topological codes based on parallel message passing between anyons, implemented as a translation-invariant cellular automaton with local feedback. By encoding inter-anyon distances in message fields and enforcing lightfront-constrained propagation, the decoder achieves a threshold for p-bounded noise and a decoding time that scales polylogarithmically with system size, notably $T_{\sf dec} = O((\log L)^{\eta})$ with $\eta$ near 1 in the toric-code setting. A multi-scale clustering/erosion analysis proves the threshold, showing that large error clusters are exponentially rare and that isolated clusters erase linearly in their size, enabling a percolation-based argument for the overall failure rate. The authors extend the construction to continuous-time Lindbladian dynamics via a marching-soldiers desynchronization, proving that the asynchronous decoder retains the same thresholds and scaling, while remaining suitable for hardware implementations. Numerics in 1D and 2D corroborate the predicted thresholds ($p_c \approx 0.5$ in 1D and $p_c \approx 7.3\%$ in 2D) and decay rates of logical errors, supporting the practicality of local, fast offline decoding for quantum memories.

Abstract

We present a local offline decoder for topological codes that operates according to a parallelized message-passing framework. The decoder works by passing messages between anyons, with the contents of received messages used to move nearby anyons towards one another. We prove the existence of a threshold, and show that in a system of linear size $L$, decoding terminates with an $O((\log L)^η)$ average-case runtime, where $η$ is a small constant. For the toric code subject to i.i.d Pauli noise, our decoder has $η=1$ and a threshold at a noise strength of $p_c\approx 7.3\%$.

Figures (20)

• Figure 1: Illustration of the message passing architecture in two dimensions. a) A schematic of how messages are produced when the decoder operates synchronously. A site hosting an anyon (green circle) sends out messages which propagate outward from the anyon's location at a constant speed (with respect to the $\infty$-norm), and increase their value by 1 at each time step. The decoder makes use of this information by moving each anyon in the direction $(\pm{\mathbf{\hat{x}}}$ or $\pm{\mathbf{\hat{y}}}$) of the received message with the smallest value (to convey directional information, the decoder actually employs four different types of messages; this distinction is not shown in the figure). b) A snapshot of the decoding dynamics in a system of size $L=75$. Darker colors indicate messages with smaller values, and the directions along which each anyon will move at the next time step are indicated by the white arrows. If the message transmission speed was infinite, each anyon would move in the direction of its nearest neighbor. For finite message speeds, it is possible for an anyon to move towards the location where an anyon was annihilated in the recent past (the rightmost anyon in the figure being an example).
• Figure 2: Clustering and error correction under the message-passing decoder, illustrated for the case of the 1D repetition code. a) A spacetime schematic illustrating how the decoder corrects errors. The noise in the input state can be recursively organized into clusters of different sizes. The smallest clusters are indicated by the locations where the blue regions intersect the $t=0$ axis. The decoder is guaranteed to eliminate these clusters before they have a chance to merge with one another, and the spacetime support of anyons contained in these small clusters is restricted to the interiors of the blue diamonds. Disregarding these small clusters, the remaining part of the noise is broken up into larger clusters (yellow regions). The decoder similarly eliminates anyons in these regions before they have a chance to merge. This clustering process is continued up to the scale of the largest clusters in the system (here, the red region), with $p_{\sf log}^{(\mathcal{E})},T_{\sf dec}^{(\mathcal{E})}$ determined by how these large clusters are eliminated. b) elimination of clusters in action. A visualization of the decoding dynamics for a system of size $L=512$, initialized on a random bit string. Black and white areas denoting regions where the bits take values of $0$ and $1$, respectively.
• Figure 3: A schematic of the message passing architecture. a) The message-passing system in 1D. Anyons (green circles) live on a 1D chain, each site of which hosts two message fields $m^\pm$. Once the messages have equilibrated, the value of $m^+_x$ at site $x$ is equal to the distance between site $x$ and the nearest anyon to the left of site $x$; in the figure, a red number $n\rightarrow$ indicates that $m^+=n$ at this site. Similarly, the value of $m^-_x$ is determined by the distance between $x$ and the nearest anyon to the right of $x$; in the figure these are indicated by the blue numbers with $\leftarrow$ arrows. With the message fields shown in the figure, the left anyon will move right, and the right anyon will move left. b) The message-passing system in 2D. Each site now hosts four message fields $m^{\pm a}$, $a = x,y$, with the figure indicating the fields created by an anyon in the indicated location. $m^{\pm a}$ fields indicate the distance (in the $\infty$ norm) from an anyon in the $\mp {\mathbf{\hat{a}}}$ direction, and propagate along the $\pm a$ section of the anyon's lightfront; in the figure, these lightfront segments are indicated by the shaded regions. The $m^{\pm a}$ field at a site $\mathbf{r}$ is updated by consulting the three sites at a infinity-norm distance of 1 in the $\mp{\mathbf{\hat{a}}}$ direction from $\mathbf{r}$. As an example, the site outlined in purple is updated according to the three sites indicated by the arrows. c) Movement of anyons in 2D. In the limit where messages are transmitted instantly, each anyon moves in the direction of its nearest neighbor (with distance calculated using the $\infty$ norm). If an anyon's nearest neighbor is located in the $\pm a$ section of that anyon's instantaneous lightfront, the anyon moves along $\pm {\mathbf{\hat{a}}}$. Thus the anyon $a$ in the figure moves along $+{\mathbf{\hat{x}}}$ towards anyon $b$, anyon $b$ moves along $+{\mathbf{\hat{y}}}$ towards anyon $c$, and so on. When the message transmission speed is finite, each anyon moves instead towards the nearest anyon in its past lightfront.
• Figure 4: lightfronts in one dimension for a message propagation speed of $v=3$; the vertical direction is time (flowing up) and the horizontal direction is space. The red points indicate (part of) ${\sf FLF}(r,t)$ with $(r,t)$ the location marked with the green point; the purple points indicate ${\sf PLF}(r,t)$.
• Figure 5: a) modified feedback rules in 2D. The arrows in each of the four quadrants indicate the direction that an anyon moves upon receiving a signal from the associated direction. When the signals are received from the $\pm {\mathbf{\hat{y}}}$ components of the lightfront, the anyon moves along $\pm {\mathbf{\hat{y}}}$ before moving along $-{\mathbf{\hat{x}}}$. Degeneracies between different movement directions are resolved using the colors of the shaded lines on the boundaries of the lightfront segments; dashed black lines mean that anyons do not move. b) strategy of the erosion proof for clusters defined with the 1 norm. Anyons initially in a ball $\mathsf{B}$ of radius $W/2$ remain there if they are sufficiently isolated. Fixing the center of the ball at $\mathbf{r}={\mathbf{0}}$, after time $t_{\sf move} = W/v$, all anyons in $\mathsf{B}$ are confined to be to the left of the line $r^1 = W/2-(t-t_{\sf move})$ (blue vertical lines); this leads to all anyons being annihilated after time $t_{\sf move} + W$.

Theorems & Definitions (74)

• Theorem 1.1: Threshold for offline decoding, informal
• Definition 2.1: synchronous automaton decoder
• Definition 2.2: $p$-bounded distributions
• Definition 3.1: message passing automaton, 1D
• Definition 3.2: lightfronts, 1D
• Definition 3.3: anyon indicator function; worldlines
• Definition 3.4: message passing automaton, general dimensions
• Definition 3.5: lightfronts, indicator functions, and worldlines: general dimensions
• Definition 4.1: clustering
• Proposition 4.1