Automorphism Ensemble Decoding of Quantum LDPC Codes

TL;DR

Quantum LDPC codes promise lower-overhead fault tolerance, but belief propagation decoders struggle due to short cycles in Tanner graphs. AutDEC introduces an automorphism-ensemble decoding framework that runs multiple BP decoders in parallel on syndrome vectors transformed by code automorphisms, selecting the most likely correction. It achieves accuracy comparable to BP-OSD-0 for $[[15,1,3]]$ Quantum Reed-Muller codes in the code-capacity setting and for Bivariate Bicycle codes under circuit-level noise, while maintaining low latency due to parallelism and without extra post-processing. The work provides open-source implementations and outlines paths to scale, including larger ensembles and generalized symmetry mappings to guide code design.

Abstract

We introduce AutDEC, a fast and accurate decoder for quantum error-correcting codes with large automorphism groups. Our decoder employs a set of automorphisms of the quantum code and an ensemble of belief propagation (BP) decoders. Each BP decoder is given a syndrome which is transformed by one of the automorphisms, and is run in parallel. For quantum codes, the accuracy of BP decoders is limited because short cycles occur in the Tanner graph and our approach mitigates this effect. We demonstrate decoding accuracy comparable to BP-OSD-0 with a lower time overhead for Quantum Reed-Muller (QRM) codes in the code capacity setting, and Bivariate Bicycle (BB) codes under circuit level noise. We provide a Python repository for use by the community and the results of our simulations.

Figures (7)

• Figure 1: Left: Original Tanner graph of the $X$ checks of QRM$[[15,1,3]]$. An error on the $15$th qubit leads to a logical error that also does not return to the codespace. Right: The permuted Tanner graph. An error on the $15$th qubit is now corrected by BP.
• Figure 2: Overview of the AutDEC online decoding phase. Upon receiving the syndrome s, $k$ paths are created. In each path, the syndrome is left-multiplied by the $r\times r$ binary linear matrix $U_{A_i}$ corresponding to the action of the automorphism $A_i$ on the stabiliser generators. The syndromes are then sent to the $k$ constituent decoders $D_{A_i}$, each of which has been pre-compiled with the permuted check matrix $HA_i$. The corrections are gathered in a list, and the most likely correction in the list is then chosen based either on the minimum weight or the priors of the detector error model.
• Figure 3: Logical error rate for the code capacity setting of the $[[15,1,3]]$ QRM code. We compare BP and BP+OSD-0 with AutBP-5. The black solid line is the $y=x$ break-even point. The shaded regions indicate the Wilson confidence interval of $95$%.
• Figure 4: Visualisation of Tanner graph of $[[72,12,6]]$$\ket{+_L}$ memory detector error model using Gephi bastian_gephi_2009.
• Figure 5: Logical error rate for different physical error rates using BP, AutBP-5, AutBP-17, AutBP-36 and BP+OSD-0 for circuit-level noise simulation of the BB [[72,12,6]] code. The BP decoders run a maximum of $1000$ iterations on a parallel schedule, with "min-sum" scaling factor of 1. The shaded regions indicate the Wilson confidence interval of 95%.

Theorems & Definitions (2)

• Example 1
• Example 2