Fig. 2: The Bell Tree.

The top figure presents a binary tree, which we call a Bell tree, with depth T = 2. Here, we assume that the single-qubit channel \({\mathcal{N}}\) consists of independent bit-flip and phase-flip channels which apply X and Z errors with probability p_x = p_z = p. The plot shows the probability of error after decoding for both the optimal decoder (orange curve) and a suboptimal but efficient decoder (blue curve), along with an upper bound on the noise threshold (violet dashed line). The latter threshold is obtained using the fact that the Bell tree has logical subtrees with branching number \(\sqrt{2}\), we show that for p > (1 − 2^−1/4)/2 ~ 8%, information always decays exponentially with depth T (see Sec. Noise Threshold for Exponential Decay of Information). Therefore, the probability of X logical error for the infinite tree is 1/2 (the same holds for the probability of logical Z errors). The blue curve corresponds to the probability of logical X error for a Bell tree of depth T = 1000 after applying an efficient decoder introduced in Sec. Recursive Decoding of Bell Tree (d = 1 code tree), which uses a recursive scheme with 2 reliability bits. The threshold for this decoder is around p_x = p_z ~ 0.5%. The orange curve corresponds to the logical X error after applying the optimal decoder for the Bell tree of depth T = 20, which is implemented via the belief propagation algorithm presented in Sec. Optimal Decoding with Belief Propagation. This plot suggests that the actual noise threshold for infinite propagation is below 1.7%.