Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes

Belief-propagation (BP) decoders play a vital role in modern coding theory, but they are not suitable to decode quantum error-correcting codes because of a unique quantum feature called error degeneracy. Inspired by an exact mapping between BP and deep neural networks, we train neural BP decoders for quantum low-density parity-check codes with a loss function tailored to error degeneracy. Training substantially improves the performance of BP decoders for all families of codes we tested and may solve the degeneracy problem which plagues the decoding of quantum low-density parity-check codes.

Figure 1

Schematics of the NBP decoder. The main cycle in the gray box is repeated $N_c$ times. Inside one cycle there are two phases of computation, the $cv\to vc$ and $vc\to cv$ phases which are governed by Eqs. (4) and (5), respectively. The inputs to the neural network are denoted by the green boxes, where the prior and syndrome correspond to $\{l_v\}$ in Eq. (4) and $\{s_c\}$ in Eq. (5), respectively. After each cycle, the set of output $\{{\mu }_{c\to v}\}$ is marginalized by Eq. (6) and the resulting $\{{\mu }_v\}$ is sent to the loss function Eq. (8). We also introduce residual connections to facilitate training of deep networks [40]. (See Supplemental Material [41] for details.)

Figure 2

Training the NBP decoder for the toric code with different code sizes. (a) The logical error rate decreases substantially after training (tested at $p_{\mathrm{err}}=0.01$). Here, the logical error rate is broken up into two terms in (b) and (d), corresponding to “flagged” and “unflagged” errors, respectively. (See main text for details.) (c) The NBP decoder exploits degeneracy by correctly decoding with an error pattern that is not exactly the same as the true error pattern. Training parameters: $N_c=25$, $lr=2\times {10}^{-4}$.

Figure 3

(Top panels) One of the observed patterns of the learned weights for the toric code. To obtain this clear pattern, we implemented weights sharing (see main text) with $\mathbf{G}=(2i,2j),i,j=0,1,\dots$. A noisy version of the same pattern is observed when weight sharing is turned off. (Bottom panels) Tanner graph of the toric code and the position of $cvvc$ weights (yellow star). Correspondence to the top panels is marked by the black box.

Figure 4

Evolution of the logical error rate as a function of the physical error rate, for the BP (NBP) decoder before (after) training. (a) Before training the performance of the BP decoder is worse for larger code sizes at all physical error rates. (b) After training, the performance improves substantially for all code sizes at all physical error rates, and the performance curves start to cross each other. This indicates the development of a threshold.

Figure 5

Training greatly improves the BP decoder for quantum LDPC codes with high rates. (a) Quantum bicycle code with code parameter $[[256,32]]$ and rate $r=0.125$. Training parameters: $N_c=12$, $lr=1\times {10}^{-4}$. (b) Quantum hypergraph-product code with code parameter $[[129,28]]$ and rate $r\sim 0.2$. Training parameters: $N_c=12$, $lr=1\times {10}^{-4}$.