Achieving Thresholds via Standalone Belief Propagation on Surface Codes

TL;DR

Novel BP decoders that exchange messages on the decoding graph and obtain code capacity thresholds via standalone BP for the surface code under depolarizing noise are proposed, applicable to any graphlike QEC code.

Abstract

The usual belief propagation (BP) decoders are, in general, exchanging local information on the Tanner graph of the quantum error-correcting (QEC) code and, in particular, are known to not have a threshold for the surface code. We propose novel BP decoders that exchange messages on the decoding graph and obtain code capacity thresholds via standalone BP for the surface code under depolarizing noise. Our approach, similarly to the minimum weight perfect matching (MWPM) decoder, is applicable to any graphlike QEC code. The thresholds observed with our decoders are close to those obtained by MWPM. This result opens the path towards scalable hardware-accelerated implementations of MWPM-compatible decoders.

Figures (6)

• Figure 1: Decoding and factor graphs. (a) Example error realization $\bm{e}$ in the $d=5$ unrotated surface code. $Z$ phase-flip errors are blue and unsatisfied syndrome plaquettes are red. (b) Decoding graph associated to $\bm{e}$. The unsatisfied syndromes are associated to red nodes and the boundary nodes are orange. We include one edge for each pair of unsatisfied syndromes and one edge that joins each unsatisfied syndrome to the boundary. Each edge has a weight $w^p \equiv (p/(1-p))^w$\ref{['eq: two-syn var factors']} and \ref{['eq: one-syn var factors']}, where $p$ is the physical error rate and $w$ is the weight of the shortest error path that satisfies the syndromes connected by the edge. We abuse the notation in this figure and refer to $w^p$ simply as $w$. Although the code has only two boundaries, we include four boundary nodes since we use each path from an unsatisfied syndrome to the boundary independently. (c) Factor graph associated to $\bm{e}$. Each unsatisfied syndrome $i$ is associated to a factor $c_i$\ref{['eq: syndrome factors']}, a factor $q_i$\ref{['eq: one-syn var factors']} and a variable $u_i$. Each pair of unsatisfied syndromes $i,j$ are associated to a factor $q_{ij}$\ref{['eq: two-syn var factors']} and a variable $v_{ij}$.
• Figure 2: Marginalization and forced convergence: (a) An example marginalization output $(\mathbb P^{(T)}(w=0), \mathbb P^{(T)}(w=1))$ for each variable $w$ and the error candidate in Fig. \ref{['fig: different factor graphs']}; (b) The error candidate provided by marginalization does not converge. (c) The error candidate provided by forced convergence, which always converges.
• Figure 3: Threshold plots. The columns determine the algorithm, and the rows are the number of iterations. The horizontal red dotted line is the pseudo-threshold (i.e. the curve where the logical error rate equals the physical error rate) and the vertical dotted line indicates the achieved threshold. Our algorithms perform best at high error-rates. All subfigures include a gray LER curve that represents pure MWPM distance 11 decoding performance. This is for comparison reasons with the corresponding distance 11 LER curve of our methods (orange). BP4M's performance seems to plateau for distances larger than 11, but we can avoid this effect by forcing the convergence and implementing BP4MF, whose decoding performance approaches MWPM.
• Figure 4: Convergence performance and MWPM postprocessing: of message-passing without forcing convergence: (a) Converged LER. (b) Convergence fraction. (c) Performance BP4M+M-$\log n$. (d) Performance BP4M+M-$\sqrt{n}$.
• Figure 5: Performance of our algorithms with $\log n$ iterations compared to MWPM for distances $d=9,11$.