# Q-Pandora Unboxed: Characterizing Resilience of Quantum Error Correction Codes Under Biased Noise
## 5 Analyzing Surface Codes Under Noise
### Surface Codes Under Various Noise Types  
Initially, we delve into how the logical error rate diminishes in response to an increase in unrotated surface code distances across a spectrum of physical error rates if the hardware is dominated by either depolarizing, gate, readout, or reset errors. For depolarizing \begin{table}
\begin{tabular}{c c c c} \hline
**QECC Type** & **Distance** & **\# Qubits** & **\# Gates** \\ \hline \multirow{4}{*}{Unrotated Surface} & 3 & 25 & 104 \\  & 5 & 81 & 328 \\  & 7 & 169 & 792 \\  & 9 & 289 & 1376 \\ \hline \multirow{4}{*}{Rotated Surface} & 3 & 17 & 64 \\  & 5 & 64 & 208 \\ \cline{1-1}  & 7 & 118 & 432 \\ \cline{1-1}  & 9 & 170 & 736 \\ \hline \end{tabular}
\end{table} 
Table 1: Characteristics of QECCs in our experiments. noise (Figure 4a), we note that increasing the distances of the unrotated surface code can effectively reduce the logical error rate, indicating the improvement in the system's error correction capabilities. Nevertheless, this improvement is only significant up to a certain point, beyond which the code distance increment fails to yield substantial benefits. This point is known as the threshold of the error-correcting code at which the physical error rate is so high that the QECC loses its efficacy, and the system becomes inherently unreliable. When rotated surface codes are being examined under depolarizing noise as shown in Figure 4e, respectively, we note similar trends such as the logical error rate reducing for higher code distances up to the threshold level. Likewise, if we prioritize the impact of gate errors as the dominant factor, the logical error rate shows similar trends when analyzed using different distances of QECCs. However, in this case, the error correction threshold is lower than the depolarizing errors. This observation holds true for unrotated (Figure 4b) and rotated (Figure 4h) surface codes. This can be explained by the fact that gate errors occur during the execution of quantum gate operations. These errors can lead to a substantial accumulation of errors over time, particularly for larger and more complex quantum circuits. In contrast, depolarizing errors are modeled as a probability that occurs before each round of computations. As such, they do not directly interfere with the gate operations themselves, and their propagation effect is thus generally lower than gate errors. The trend of logical error rate reduction with increasing unrotated surface code distance applies to readout and reset errors as well, as depicted in Figure 4c,d. However, the key distinction lies in the higher thresholds of readout and reset errors as compared to both depolarizing and gate errors. This can be attributed to the lack of propagation effects associated with readout and reset errors, which are primarily linked with the measurement and initialization processes in quantum computing. To be more specific, readout and reset errors, despite producing incorrect results, do not accumulate. All of these characteristics hold true for rotated surface codes (Figure 4g,h). For further clarification, we have marked the general error rate of a state-of-the-art quantum processor, which is approximately \(10^{-3}\), in all the plots in Figure 4 using a horizontal and a vertical black dotted line. These lines indicate that both the physical and logical error rates would be situated at or below this level without the application of  
Figure 4: Logical error rate vs. physical error rate in unrotated and rotated surface codes with varying distances for various error types. The black dotted line marks the \(10^{-3}\) error rate of modern processors. Errors include (**a**) depolarizing, (**b**) gate, (**c**) reset, and (**d**) readout for unrotated surface codes, with (**e–h**) representing similar experiments with rotated surface codes. quantum error correction. By applying a QECC, we can significantly decrease the logical error rate below this level, and as the distance of the codes is increased, the system will become more fault tolerant.