Tensor-Network Simulations of the Surface Code under Realistic Noise

The surface code is a many-body quantum system, and simulating it in generic conditions is computationally hard. While the surface code is believed to have a high threshold, the numerical simulations used to establish this threshold are based on simplified noise models. We present a tensor-network algorithm for simulating error correction with the surface code under arbitrary local noise. We use this algorithm to study the threshold and the subthreshold behavior of the amplitude damping and systematic rotation channels. We also compare these results to those obtained by making standard approximations to the noise models.

Figure 1

Simulation of error correction with the surface code. (a) The surface-code density operator as a tensor network. The two additional indices on the bottom right tensor are ancilla indices. (b) Applying local noise to the state. (c) Applying a check projector $P_k$ (red) to the state. Green tensors have been involved in previous check measurements, while blue tensors have not. (d) The resulting tensor network representing $\mathrm{Tr}(P_{k+1}{\rho }_k)$ obtained by capping off pairs of physical indices.

Figure 2

Empirical logical error rate vs lattice dimension for two non-Pauli noise models. Solid black lines represent exact channels, while dashed and dotted lines represent the PTA and the HPA, respectively. Plot (a) depicts $\gamma =9\%$ amplitude damping, while plot (b) depicts $\theta =0.005\pi$ systematic rotation. For each set of parameters, $1.32\times 10^5$ syndromes were sampled to acquire statistics. For the PTA of SR with $W>5$, error rates are numerically zero. Error rates obtained for the exact channels using an approximate contraction algorithm with $\chi =8$ are also plotted in blue. Note that the exact average logical error could significantly differ from the empirical average observed over millions of syndrome measurements if failures are dominated by outliers.