Optimal Decoding with the Worm

TL;DR

A new decoder for ``matchable''qLDPC codes that uses a Markov-Chain Monte-Carlo algorithm to approximately compute the probabilities of logical error classes given a syndrome is proposed, and it is found that the threshold for ``correlated worm decoding'' is substantially higher than for both minimum-weight perfect matching and for correlated matching.

Abstract

We propose a new decoder for ``matchable'' qLDPC codes that uses a Markov-Chain Monte-Carlo algorithm -- called the \emph{worm algorithm} -- to approximately compute the probabilities of logical error classes given a syndrome. The algorithm hence performs (approximate) \emph{optimal} decoding, and we expect it to be computationally efficient in certain settings. The algorithm is applicable to decoding random errors for the surface code, the honeycomb Floquet code, and hyperbolic surface codes with constant rate, in all cases with and without measurement errors. The efficiency of the decoder hinges on the mixing time of the underlying Markov chain. We give a rigorous mixing time guarantee in terms of a quantity that we call the \emph{defect susceptibility}. We connect this quantity to the notion of disorder operators in statistical mechanics and use this to argue (non-rigorously) that the algorithm is efficient for \emph{typical} errors in the entire decodable phase. We also demonstrate the effectiveness of the worm decoder numerically by applying it to the surface code with measurement errors as well as a family of hyperbolic surface codes. For most codes, the matchability condition restricts direct application of our decoder to noise models with independent bit-flip, phase-flip, and measurement errors. However, our decoder returns \emph{soft information} which makes it useful also in heuristic ``correlated decoding'' schemes which work beyond this simple setting. We demonstrate this by simulating decoding of the surface code under depolarizing noise, and we find that the threshold for ``correlated worm decoding'' is substantially higher than for both minimum-weight perfect matching and for correlated matching.

Figures (14)

• Figure 1: Schematic illustration of optimal decoding using the worm algorithm. A reference matching consistent with the observed syndrome is first identified and used to reweight the decoding graph. The worm algorithm then samples closed-loop configurations on the reweighted graph, thereby generating error chains consistent with the syndrome. These samples can be used to determine the most likely logical class of errors, enabling optimal decoding.
• Figure 2: Illustration of the addition of an effective boundary detector for the DEM of the surface code with i.i.d. bit-flip noise. Red circles correspond to detectors (stabilizers), edges correspond to error mechanisms (qubits) and the blue circle correspond to the additional boundary node added. Note that both the left and right blue circles correspond to the same boundary node.
• Figure 3: Schematic representation of the worm process. A pair of virtual defects is created at a random vertex. One performs a random walk on the graph until it meets the other, forming a loop update. Red crosses denote the observed syndrome.
• Figure 4: Schematic of the worm-decoding procedure. The logical sector of each sampled loop configuration is measured on the decoding graph of the toric code (here without measurement errors), and the corresponding logical tally is updated.
• Figure 5: Schematic illustrating how a pair of virtual defects (blue crosses) can form near a reference matching (red dashed line) of length $\ell$, and subsequently become trapped in the vicinity of the syndrome locations (red crosses). While such configurations of size $\ell$ contribute a mixing time that grows exponentially in $\ell$, they are also exponentially rare in $\ell$ for local random error models.

Theorems & Definitions (12)

• Theorem 4.2: short version
• Proposition D.2: Andreas Galanis, private communication
• proof
• Theorem E.1: Schweinsberg (2002) schweinsberg2002relaxation_time
• Theorem E.2
• Lemma E.3
• proof
• Claim E.4
• proof
• Claim E.5