Local decoder for the toric code with a high pseudo-threshold

Abstract

Local decoders provide a promising approach to real-time quantum error-correction by replacing centralized classical processing, with significant hardware constraints, by a fully distributed architecture based on a simple, local update rule. We propose a new local decoder for Kitaev's toric code: the 2D signal-rule, that interprets odd parity stabilizer measurements as defects, attracted to each other via the exchange of binary signals. We present numerical evidence of exponential logical error suppression with system size below some critical error rate, under a phenomenological noise model, with data and measurement errors between each iteration. Compared to previously known local decoders, which exhibit suboptimal thresholds and scaling, our construction halves (in log scale) the threshold gap with state-of-the-art decoders, and achieves optimal scaling for experimentally relevant system sizes, enabling the practical realization of a two-dimensional local quantum memory.

Figures (5)

• Figure 1: Layout of the 2D signal-rule. (a) Each $Z$ stabilizer site hosts a classical processor storing the stabilizer measurement outcome (i.e., the defect) together with 1- and 2-forward signals, anti-signals, and stacks for each cardinal direction. Directions unused by the rules (in grey) are retained to define rules by symmetry. (b) Elementary rules of the decoder: filled symbols indicate particles, colored contours indicate empty sites, and plus/minus signs denote updates, subject to implicit constraints from the state space (e.g., no signal creation on an occupied site). A circular arrow marks fourfold rotational symmetry, while $\pi$ denotes symmetry between opposite directions. ($i$) Matching neighboring defects via a Pauli correction (in light blue). ($ii$) Emission of 1-forward-signals from a defect, and associated 1-stack increment. ($iii$) Emission of 2-forward-signals from a 1-forward-signal or a defect, and associated 2-stack increment. ($iv$) Displacement of a defect receiving a forward signal. ($v$) Emission of 1-anti-signals from the decrement of a non-empty 1-stack in the absence of a defect. ($vi$) Emission of 2-anti-signals from the decrement of a non-empty 2-stack in the absence of a defect and 1-forward-signal. ($vii$) Recombination of a forward-signal with an anti-signal of the same type and direction. ($viii$) Propagation of 1-forward-signals; other signals propagate analogously.
• Figure 2: Defect attraction via forward signal propagation. (a) Attractive interaction generated by a single isolated defect, mediated by forward signals; stacks and anti-signals are omitted for clarity. At each iteration of the decoder, 1-forward-signals (in dark blue) are emitted from the defect and propagate outward, and subsequently 2-forward-signals (in light blue) are emitted from the 1-forward-signals and defects and propagate further outward, generating a 2-dimensional front wave. (b) Attraction basin of a defect, with arrows indicating the direction of attraction for another defect at each site. (c) The assymmetry of the attraction bassin ensures agreement between defects displacements during the matching process.
• Figure 3: Dynamics following a single initial measurement error. (a) Forward signals and defects are displayed in the first column, while stacks and anti-signals are shown together in the second column and separated by cardinal direction in the four remaining columns. Since only one particle is drawn per site in the second column, particles are displayed according to the following priority order: 1-stacks, 2-stacks, 1-anti-signals, and 2-anti-signals. The maximum stack value over the four cardinal directions is indicated by a white number. (b) Representation of a sequence of configurations of the 2D signal-rule decoder, where steps leaving the configuration unchanged are not shown. In the presence of an initial measurement error, the increment of the stacks upon emission of forward-signals ensure that all signals initially sent eventually recombine at the end.
• Figure 4: Performance of the $2$D signal-rule decoder. (a) Logical error rate as a function of the physical error probability $\varepsilon$ for a phenomenological noise model with $\varepsilon=\varepsilon_d=\varepsilon_m$, shown for several code distances $d$. The logical error rate is obtained by normalizing the failure rate measured in Monte Carlo simulations. The data are fitted using the ansatz $\tfrac{A}{d} (\varepsilon/\varepsilon_c)^{\gamma_d}$, with a distinct exponent $\gamma_d$ for each code distance, which gives $\varepsilon_c = 0.68\%$. (b) Effective distance $\gamma_d$ as a function of the code distance $d$, for different stack upper bounds. The optimal scaling in $\tfrac{d+1}{2}$, e.g. reached with minimum-weight perfect matching decoding, is indicated by the black dashed line. (c) For each successive pair of distances $d_i < d_{i+1}$, we fit the modified ansatz $\tilde{A}(\varepsilon / \tilde{\varepsilon}_c)^{\tilde{\gamma}}$ to the corresponding restricted data. This yields a unique crossing point $\tilde{\varepsilon}_c[d_i] = f[d_i, d_{i+1}]$, plotted as a function of $d_i$, and compared to the critical value $\varepsilon_c$ (obtained for all $d$), shown as a dotted line.
• Figure 5: Markovian dynamics. (a) $1 - \frac{4}{3} P_L(\tau)$ as a function of the simulation time $\tau$; a constant slope on a logarithmic scale indicates a logical flip probability that is independent of the total simulation time. The logical error rate $\varepsilon_L$ is extracted from the asymptotic regime in which $P_L(\tau)/\tau$ reaches a constant value. The convergence time to this asymptotic regime is estimated using simulations over shorter time scales, by determining when $P_L(\tau)/\tau$ approaches $\varepsilon_L$. We define the convergence time as $\tau_d := \min \bigl\{ \tau_0 \ge 0 \,|\, P_L(\tau)/\tau > 0.9 \cdot \varepsilon_L,\ \forall\, \tau \ge \tau_0 \bigr\}$, which is shown in panel (c).