## 5 Numerical Results for Specific Error Models  
### Uncorrelated Errors  
When benchmarking the planar code for a given decoder, it is typical to start with the case in which syndrome measurements are noiseless, and so all noise is due to errors acting on the spins of the code. Let us consider a general Pauli channel acting independently on each spin of the code, applying \(\sigma_{\alpha}\) with probability \(p_{\alpha}\) for \(\alpha\in\{x,y,x\}\). Optimal decoding will fully take into account the exact values of these probabilities, and any correlations between the plaquette and vertex syndromes that they will cause. However, the decoder considered here takes the simpler route in which the two syndrome types are decoded independently. The plaquette decoding then depends only on the probability \(p=p_{x}+p_{y}\) with which a bit flip, either \(\sigma_{x}\) or \(\sigma_{y}\), is applied. The vertex decoding similarly depends only on the probability \(\tilde{p}=p_{z}+p_{y}\) for phase flips. Without loss of generality we therefore consider the error model in which bit and phase flips occur independently with the probabilities \(p\) and \(\tilde{p}\), respectively. We assume that \(p>\tilde{p}\), so that we consider the worst of the two decoding problems. The decoder is benchmarked using the logical error rate, \(P\), that it achieves for different bit flip error rates, \(p\), and linear system sizes, \(L\). This is done by randomly generating error configurations for the qubits of the code according to the noise model, applying the decoder to the resulting anyons, and then determining whether or not a logical error occurred for each sample. For each case we use the number of samples, \(n\), required in order for \(10^{3}\) logical errors to occur. The logical error rate is then \(P=10^{3}/n\). The data was obtained to determine :  
1. The threshold error rate, \(p_{c}\), for which \(P\) decays with \(L\) when \(p<p_{c}\);
2. The minimum system size, \(L^{*}\), required for each \(p\) such that \(P<p\) (and so the minimum system size required for a code stored qubit to be less noisy than one stored on a single physical spin);
3. The decay of the logical error rate for bit flip error rates well below threshold, to show that effective error correction occurs;
4. The values of \(\alpha(p)\) for each \(p\), when the above data is fitted to a function \(P=0(e^{-\alpha(p)(cL)^{\beta}})\). The results can all be found in Figure 2. These suggest that the threshold is around \(7.25\%\)-\(7.5\%\). This is of the same order as the upper bound on the threshold of around \(11\%\). It also compares well to the threshold of \(7.8\%\) obtained by the renormalization decoder of [17]. It is larger than the \(6.7\%\) threshold obtained by the Bravyi-Haah algorithm [3], but is less than the \(8.4\%\) threshold of the modified version in [4]. Note that these decoders deal with the toric code, which is identical to the planar code in the bulk but differs in its boundary conditions. Since the planar code boundary is known to lead to worse decoding behaviour [18], it is remarkable that this decoder compares so well in its performance. It is found that logical error rates of \(P<p\) can be obtained using a small code of size \(L=4\) for all \(p\lesssim p_{c}/2\). The decoder therefore does not require a great deal of physical qubits in order to get good results. However, the required system size rises steeply for higher \(p\). For \(p\lesssim p_{c}/2\) we also find that the logical error rate decays very quickly as \(L\to\infty\). The decay below threshold will be of the form \(P=0(e^{-\alpha(p)(cL)^{\beta}})\). Here the scaling can be expected to be governed by the minimum requirements for a logical error. This means \(\beta=\log_{3}2\approx 0.63\) and, to be conservative, \(c=1/4\). To test this assumption, we can consider the scaling of \(\log(-\log P)\) against \(\log L\). This should be a linear relationship, with \(\beta\) as the gradient. To minimize finite size effects, it is best to study low values of \(p\).