Fast erasure decoder for hypergraph product codes

Abstract

We propose a decoder for the correction of erasures with hypergraph product codes, which form one of the most popular families of quantum LDPC codes. Our numerical simulations show that this decoder provides a close approximation of the maximum likelihood decoder that can be implemented in O(N^2) bit operations where N is the length of the quantum code. A probabilistic version of this decoder can be implemented in O(N^1.5) bit operations.

Figures (6)

• Figure 1: Two examples of an erasure-induced subgraph for a simple Tanner graph $T(H)$. Non-erased nodes are grayed-out and excluded from the subgraph.
• Figure 2: The HGP code derived from a linear code with 7 bits and 3 checks. The support of the $Z$ stabilizer generator with index $(b, a') \in B \times A$ is given by the neighbors of $(b, a')$ in the Cartesian product of the graph $T(H)$ with itself. In the product notation, we follow the $x\times y$ convention, where the first coordinate denotes the horizontal code and the second coordinate denotes the vertical code.
• Figure 3: Example of a stabilizer stopping set for the distance 3 surface code obtained from a hypergraph product of two 3-bit repetition codes. The qubits in the support of an $X$-stabilizer are a stopping set for $T(H_Z)$. Highlighted nodes show the erasure subgraph corresponding to this stopping set.
• Figure 4: Performance of the pruned peeling and VH decoders using four HGP codes and compared with the ML decoder ($10^6$ simulations per data point). Plots show the failure rates of the decoders for recovering an X-type Pauli error supported on the erasure vector, up to multiplication by a stabilizer. In these plots, we discard the data points corresponding to fewer than 20 decoding failures to show only meaningful data. All the data points with at least 20 failures are shown. Note that the curves for $M=1$ and $M=2$ virtually overlap in all cases.
• Figure 5: (a) Example showing how the VH graph is computed from the erasure subgraph (subgraph edges are excluded for simplicity). Nodes in the VH graph correspond to clusters of erased qubits (erased qubit-nodes in the same row or column sharing a check-node, indicated by a blue box above). Edges in the VH graph correspond to connecting checks (check nodes adjacent to both a vertical and horizontal cluster, indicated by a red box above). Isolated clusters have no connecting checks (degree 0 nodes in the VH graph). Dangling clusters have exactly one connecting check (degree 1 nodes in the VH graph). A dangling cluster is determined to be frozen or free based on the connectivity of the subgraph induced by the cluster; (b) and (c) show two examples for the vertical dangling cluster in the purple box with different subgraphs. (b) An example of a frozen cluster. All solutions for the cluster which satisfy the internal checks have the same contribution to the connecting check. (c) An example of a free cluster. There exist solutions to the internal checks of the cluster which are 0 or 1 on the connecting check; equivalently, there exists an error on the cluster whose syndrome is non-zero only on the single connecting check.

Theorems & Definitions (6)

• Lemma 1: Stabilizer stopping sets
• proof
• Lemma 2: Horizontal and vertical stopping sets
• proof
• Proposition 1
• proof