Modified belief propagation decoders for quantum low-density parity-check codes

Quantum low-density parity-check codes can be decoded using a syndrome based GF(4) belief propagation decoder (where GF denotes Galois field). However, the performance of this decoder is limited both by unavoidable 4-cycles in the code's factor graph and the degenerate nature of quantum errors. For the subclass of CSS codes, the number of 4-cycles can be reduced by breaking an error into an $X$ and $Z$ component and decoding each with an individual $\mathrm{GF}\left(2\right)$ based decoder. However, this comes at the expense of ignoring potential correlations between these two error components. We present a number of modified belief propagation decoders that address these issues. We propose a $\mathrm{GF}\left(2\right)$ based decoder for CSS codes that reintroduces error correlations by reattempting decoding with adjusted error probabilities. We also propose the use of an augmented decoder, which has previously been suggested for classical binary low-density parity-check codes. This decoder iteratively reattempts decoding on factor graphs that have a subset of their check nodes duplicated. The augmented decoder can be based on a $\mathrm{GF}\left(4\right)$ decoder for any code, a $\mathrm{GF}\left(2\right)$ decoder for CSS code, or even a supernode decoder for a dual-containing CSS code. For CSS codes, we further propose a $\mathrm{GF}\left(2\right)$ based decoder that combines the augmented decoder with error probability adjustment. We demonstrate the performance of these new decoders on a range of different codes, showing that they perform favorably compared to other decoders presented in literature.

Figure 1

The factor graph of the [7,4,3] Hamming code corresponding to the parity-check matrix given in Eq. (12). Error nodes are represented as circles and check nodes as squares.

Figure 2

The effect of augmentation density and random perturbation strength on decoder performance for the [[400,200]] bicycle code on the depolarizing channel. Each decoder uses $N=10$ maximum attempts.

Figure 3

FER performance of decoders with $N=100$ attempts (where applicable) for the [[400,200]] bicycle code on the depolarizing channel.

Figure 4

Average number of iterations required by decoders with $N=100$ attempts (where applicable) for the [[400,200]] bicycle code on the depolarizing channel.

Figure 5

FER performance of decoders at $p=0.008$ with a varying number of decoding attempts for the [[400,200]] bicycle code on the depolarizing channel.

Figure 6

The effect of augmentation density and random perturbation strength on decoder performance for the [[400,200]] bicycle code on the $XZ$ channel. Each decoder uses $N=10$ maximum attempts.

Figure 7

FER performance of decoders with $N=100$ attempts (where applicable) for the [[400,200]] bicycle code on the $XZ$ channel.

Figure 8

Average number of iterations required by decoders with $N=100$ attempts (where applicable) for the [[400,200]] bicycle code on the $XZ$ channel.

Figure 9

FER performance of decoders at $p=0.008$ with a varying number of decoding attempts for the [[400,200]] bicycle code on the $XZ$ channel.

Figure 10

The effect of augmentation density and random perturbation strength on decoder performance for the [[610,490]] BIBD code on the depolarizing channel. Each decoder uses $N=10$ maximum attempts.

Figure 11

FER performance of decoders with $N=100$ attempts (where applicable) for the [[610,490]] BIBD code on the depolarizing channel.

Figure 12

Average number of iterations required by decoders with $N=100$ attempts (where applicable) for the [[610,490]] BIBD code on the depolarizing channel.

Figure 13

FER performance of decoders at $p=0.001$ with a varying number of decoding attempts for the [[610,490]] BIBD code on the depolarizing channel.

Figure 14

The effect of augmentation density and random perturbation strength on decoder performance for the [[506,240]] quasicyclic code on the depolarizing channel. Each decoder uses $N=10$ maximum attempts.

Figure 15

FER performance of decoders with $N=100$ attempts (where applicable) for the [[506,240]] quasicyclic code on the depolarizing channel.

Figure 16

Average number of iterations required by decoders with $N=100$ attempts (where applicable) for the [[506,240]] quasicyclic code on the depolarizing channel.

Figure 17

FER performance of decoders at $p=0.015$ with a varying number of decoding attempts for the [[506,240]] quasicyclic code on the depolarizing channel.

Figure 18

The effect of augmentation density and random perturbation strength on decoder performance for the [[400,200]] bicyclelike code on the depolarizing channel. Each decoder uses $N=10$ maximum attempts.

Figure 19

FER performance of decoders with $N=100$ attempts (where applicable) for the [[400,200]] bicyclelike code on the depolarizing channel.

Figure 20

Average number of iterations required by decoders with $N=100$ attempts (where applicable) for the [[400,200]] bicyclelike code on the depolarizing channel.

Figure 21

FER performance of decoders at $p=0.012$ with a varying number of decoding attempts for the [[400,200]] bicyclelike code on the depolarizing channel.

Figure 22

The effect of augmentation density and random perturbation strength on decoder performance for the [[400,202]] non-CSS code $A$ on the depolarizing channel. Each decoder uses $N=10$ maximum attempts.

Figure 23

FER performance of decoders with $N=100$ attempts (where applicable) for the [[400,202]] non-CSS code $A$ on the depolarizing channel.

Figure 24

Average number of iterations required by decoders with $N=100$ attempts (where applicable) for the [[400,202]] non-CSS code $A$ on the depolarizing channel.

Figure 25

FER performance of decoders at $p=0.012$ with a varying number of decoding attempts for the [[400,202]] non-CSS code $A$ on the depolarizing channel.

Figure 26

The effect of augmentation density and random perturbation strength on decoder performance for the [[400,201]] non-CSS code $B$ on the depolarizing channel. Each decoder uses $N=10$ maximum attempts.

Figure 27

FER performance of decoders with $N=100$ attempts (where applicable) for the [[400,201]] non-CSS code $B$ on the depolarizing channel.

Figure 28

Average number of iterations required by decoders with $N=100$ attempts (where applicable) for the [[400,201]] non-CSS code $B$ on the depolarizing channel.

Figure 29

FER performance of decoders at $p=0.012$ with a varying number of decoding attempts for the [[400,201]] non-CSS code $B$ on the depolarizing channel.