Linear-time general decoding algorithm for the surface code

A quantum error correcting protocol can be substantially improved by taking into account features of the physical noise process. We present an efficient decoder for the surface code which can account for general noise features, including coherences and correlations. We demonstrate that the decoder significantly outperforms the conventional matching algorithm on a variety of noise models, including non-Pauli noise and spatially correlated noise. The algorithm is based on an approximate calculation of the logical channel using a tensor-network description of the noisy state.

Figure 1

Layout of the surface code where qubits are located at vertices. Orange faces represent $A_f$ checks and white faces represent $B_f$ checks. The logical $\overline{Z}$ operator is a product of $Z$ along the dashed blue and the logical $\overline{X}$ operator is a product of $X$ along the dashed red line.

Figure 2

Decoder performance under high-strength amplitude damping (a), low-strength amplitude damping (b), and correlated bit-flip noise (c). For amplitude damping, the logical error rate is expressed in terms of the diamond distance of the logical channel from the identity, which we have computed using our surface-code simulation algorithm [25]. In (a), error rates are close to optimal threshold ($\gamma =39\pm 1\%$) [25], and $\gamma$ is varied. In (b), a low error rate $\gamma =9\%$ is fixed, and the lattice width $W$ is varied. In both cases, an asymmetric lattice with length $2W-1$ (i.e., longer logical $\overline{X}$) was used because amplitude damping has a greater tendency to flip $z$ checks than $x$ checks. For correlated bit-flip noise (c), the $y$ axis is the probability of a logical error, which is estimated by sampling over physical error configurations. For the TN decoder, no logical errors were observed for $1/\beta \le 0.7$, implying logical error rates of less than $3\times {10}^{-5}$.