Local active error correction from simulated confinement

TL;DR

The paper presents a fully local, self-organized cellular-automaton decoder that simulates confinement between spacetime defects to enable fault-tolerant error correction in topological codes. By adding a polylogarithmic-depth RG-like dimension $Z$, defects are effectively renormalized as they propagate, yielding a threshold with memory time scaling $t_{\mathrm{mem}} = (p_c/p)^{\Omega(L^\beta)}$. Numerics on the 2D surface code under depolarizing and measurement noise show a threshold near $p_c \approx 1.5\%$ for synchronous updates, dropping under asynchronous updates, while analytic results prove threshold existence under appropriate $Z$ and update velocity constraints. The work also analyzes desynchronization, Gerrymandering effects, initialization, and phase transitions, providing a comprehensive assessment of local, real-time decoding with practical resource estimates and clear directions for future improvements. Overall, this approach advances fault-tolerant QEC by achieving a provable threshold with fully local operations and scalable memory lifetimes, potentially informing hardware implementations of real-time quantum error correction.

Abstract

We refine an old idea for performing fault-tolerant error correction in topological codes by simulating confining interactions between excitations. We implement confinement using an array of local classical processors that measure syndromes, broadcast messages to neighboring processors, and move excitations using received messages. The dynamics of the resulting real-time decoder is geometrically local, homogeneous in spacetime, and self-organized, operating without any form of global control. We prove that below a threshold error rate, it achieves a memory lifetime scaling as a stretched exponential in the linear system size $L$, provided that it has access to $O({\rm polylog}(L))$ noiseless classical bits for each noisy qubit. When applied to the surface code subject to depolarizing noise and measurement errors of equal strength, numerics indicate a threshold at $p_c \approx 1.5\%$.

Figures (11)

• Figure 1: A summary of the message-passing architecture that simulates confinement. $\mathsf{a})$ A schematic of a $2d$ CA decoder. Qubits (red balls) live on the links of a square lattice, and each lattice site $\mathbf{r}$ hosts a classical processor $\mathsf{P}_\mathbf{r}$. The classical system performs error correction through a combination of local measurements, local feedback, and communication between neighboring processors. In our decoder, each $\mathsf{P}_\mathbf{r}$ stores information about defects at $\mathbf{r}$ ($s_\mathbf{r}$) and messages that are used to generate confining interactions between defects ($m_\mathbf{r}$). $\mathsf{b})$ The exchange of messages between defects (green balls) leads to confining interactions. The strength of the messages emitted by defects is indicated by the purple shading, and feedback is applied to move defects in the direction of the strongest received signal (white arrows). ${\sf c)}$ A schematic of how defects produce messages. In $d$ dimensions, each defect produces multiple types of messages, one for each (positive and negative) coordinate direction. Messages emitted from $\mathbf{r}$ along the $\pm {\mathbf{\hat{a}}}$ direction spread to sites $\mathbf{r}'$ such that $r'^a-r^a \gtreqless r'^b- r^b$ for all $b=1,\dots,d$, and decrease their strength as they do so, with stronger messages overwriting weaker ones (and with degeneracies along the diagonals being lifted according to the convention in the figure).
• Figure 2: A schematic of the decoder operation and error-correction process, shown in one spatial dimension for simplicity of illustration. Qubits live on the links of the $1d$ chain indicated in blue, and the lattice above this chain (indicated as the classical control region) contains the internal variables stored by the classical processors $\mathsf{P}_\mathbf{r}$ at each site $\mathbf{r}$ (an example of the internal variables controlled by a particular processor is indicated by the green shaded region). The processors store a lattice of classical variables of depth $Z$, and we prove that a threshold is present as long as $Z = \Omega({\rm polylog}(L))$. Syndrome-changing events ("defects") are marked as green circles, and move "upwards" in the classical control region under the decoding dynamics until accumulating on the back wall. During this process, defects use the message-passing scheme to attempt to pair-annihilate; the white arrows indicate the directions that the anyons would move in the limit of instantaneous message speeds. The section labeled future error history illustrates errors in spacetime that have not yet been experienced by the system; as time increases these errors are "fed into" the quantum system, and then propagate up along the $Z$ direction. The regions marked $C_0$ and $C_1$ indicate noise clusters of varying spacetime support. We show that only clusters of linear size $\gtrsim Z$ can survive to the back wall, where they are then corrected unless they coagulate into a larger cluster of size $\sim L$.
• Figure 3: A schematic illustrating why $Z=0$ message-passing decoders and field-based decoders have a pseudothreshold. For noise of strength $p$, the typical distance between anyon pairs created by the noise is $\xi_p \sim p^{-1/d}$. When a pair of anyons $a_1,a_2$ at $\mathbf{r}_1,\mathbf{r}_2$ is created with separation $||\mathbf{r}_1-\mathbf{r}_2||\gg \xi_p$, the intervening small pairs created by the noise will screen the interaction between $a_1$ and $a_2$, leading to these anyons randomly diffusing and escaping correction. Since the time scale for the noise to create a pair of separation $r$ is exponentially long in $r$, this argument gives a memory lifetime scaling as $t_{\sf mem} \sim e^{\xi_p}$.
• Figure 4: Pseudothreshold behavior in $t_{\sf mem}$ for message-passing decoders with $Z=0$, shown for the $1d$ repetition code (left) and the $2d$ toric code (right).
• Figure 5: A schematic of the error model used in the proof of theorem \ref{['thm:nogo']}. The noise is restricted to create well-separated nearest-neighbor anyons at reflection-symmetric locations on the "outside" of any large pair.

Theorems & Definitions (27)

• Theorem 1: existence of a threshold, informal
• Definition 1: clustering
• Definition 2: noise hierarchies and level-$k$ error rates
• Definition 3: memory time
• Theorem 2: pseudothresholds for historyless message passing
• Theorem 3: existence of a threshold
• Lemma 1
• Definition 4: clumps
• Lemma 2
• proof