Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities

Quantum 9, 1950 (2025).

https://doi.org/10.22331/q-2025-12-16-1950

We analyze the performance of quantum stabilizer codes, one of the most important classes for practical implementations, on both symmetric and asymmetric quantum channels. To this aim, we first derive the weight enumerator (WE) for the undetectable errors based on the quantum MacWilliams identities. The WE is then used to evaluate tight upper bounds on the error rate of CSS quantum codes with minimum weight decoding. For surface codes we also derive a simple closed form expression of the bounds over the depolarizing channel. We introduce a novel approach that combines the knowledge of WE with a logical operator analysis, allowing the derivation of the exact asymptotic error rate for short codes. For example, on a depolarizing channel with physical error rate $rho to 0$, the logical error rate $rho_mathrm{L}$ is asymptotically $rho_mathrm{L} approx 16 rho^2$ for the $[[9,1,3]]$ Shor code, $rho_mathrm{L} approx 16.3 rho^2$ for the $[[7,1,3]]$ Steane code, $rho_mathrm{L} approx 18.7 rho^2$ for the $[[13,1,3]]$ surface code, and $rho_mathrm{L} approx 149.3 rho^3$ for the $[[41,1,5]]$ surface code. For larger codes our bound provides $rho_mathrm{L} approx 1215 rho^4$ and $rho_mathrm{L} approx 663 rho^5$ for the $[[85,1,7]]$ and the $[[181,1,10]]$ surface codes, respectively. Finally, we extend our analysis to include realistic, noisy syndrome extraction circuits by modeling error propagation throughout gadgets. This enables estimation of logical error rates under faulty measurements. The performance analysis serves as a design tool for developing fault-tolerant quantum systems by guiding the selection of quantum codes based on their error correction capability. Additionally, it offers a novel perspective on quantum degeneracy, showing it represents the fraction of non-correctable error patterns shared by multiple logical operators.