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

TL;DR

This work develops a MacWilliams-identity–based framework to analyze the performance of stabilizer quantum codes, including CSS and surface codes, on symmetric and asymmetric channels. By deriving the undetectable-errors weight enumerator and connecting it to logical operators, the authors obtain tight BD and MW-decoding bounds, and, for surface codes, closed-form WE expressions that yield exact asymptotic logical-error rates for short codes. The analysis is extended to realistic, noisy syndrome extraction via cat states, flag gadgets, and Steane gadgets, enabling circuit-level error-rate estimation and bounding of logical failures under practical fault-tolerance scenarios. Additionally, the paper offers a fresh perspective on quantum degeneracy by linking it to the sharing of error patterns among logical operators, guiding code design. Together, these results provide a concrete design tool for choosing QECCs and understanding degeneracy effects in fault-tolerant quantum systems, with explicit examples for Shor, Steane, and various surface codes.

Abstract

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 \acl{MW} 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 $ρ\to 0$, the logical error rate $ρ_\mathrm{L}$ is asymptotically $ρ_\mathrm{L} \approx 16 ρ^2$ for the $[[9,1,3]]$ Shor code, $ρ_\mathrm{L} \approx 16.3 ρ^2$ for the $[[7,1,3]]$ Steane code, $ρ_\mathrm{L} \approx 18.7 ρ^2$ for the $[[13,1,3]]$ surface code, and $ρ_\mathrm{L} \approx 149.3 ρ^3$ for the $[[41,1,5]]$ surface code. For larger codes our bound provides $ρ_\mathrm{L} \approx 1215 ρ^4$ and $ρ_\mathrm{L} \approx 663 ρ^5$ 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.

Figures (10)

• Figure 1: $[[ 13,1,3 ]]$ Surface code. (a) Actual lattice with data qubits $D$ (circles) and ancillae $A$ (squares). (b) Simplified representation with $\bm{X}$ generators corresponding to sites and $\bm{Z}$ generators to plaquettes. A smooth edge ($A_2$) and a rough edge ($A_{10}$) are depicted in green and blue, respectively. Examples of logical operators are drawn on the lattice.
• Figure 2: Example of errors leading to a logical operator of $w=4$ for the $[[ 13,1,3 ]]$ surface code. $\bm{Z}$, $\bm{X}$, and $\bm{Y}$ errors on qubits are depicted in red, purple, and blue, respectively. (a) $\bm{Z} _L$ occurs if $\bm{Z}$ correction operators are applied on data qubits $D_5$ and $D_6$. The errors are corrected if the MWPM decoder applies $\bm{Z}$ on $D_3$ and $D_7$. (b) $\bm{Z} _L$ occurs if $\bm{Z}$ correction operators are applied on $D_3$ and $D_7$. The errors are corrected if the MWPM decoder applies $\bm{Z}$ on $D_5$ and $D_6$. (c) $\bm{Z} _L$ occurs if $\bm{Z}$ correction operators are applied on $D_3$ and $D_6$. The MWPM decoder could apply also $\bm{Z}$ operators on $D_5$ and $D_7$, correcting the error, or on $D_2$ and $D_4$, correcting the error. (d) $\bm{Z} _L$ occurs if $\bm{Z}$ correction operators are applied on $D_5$ and $D_7$. The MWPM decoder could apply also $\bm{Z}$ operators on $D_3$ and $D_6$ or on $D_2$ and $D_4$, causing a different logical operator. ($a^\prime, c^\prime$) Analogous examples for $\bm{X} _L$ logical operator. ($e, f$) Logical operators of $w = 4$ caused by $\bm{Y} \bm{Y}$ errors.
• Figure 3: Logical error rate, $[[9,1,3]]$ Shor code over a depolarizing channel. Comparison between theoretical analysis (curves) and simulation (symbols). The curves refer to: the BD decoding performance \ref{['eq:PeGen']}; the MW decoding upper bound \ref{['eq:error_probWithBetaUB']} and its asymptotic approximation \ref{['eq:error_probWithBetaApprox']} with the exact $\beta_2$ from Tab. \ref{['tab:paramBeta']}.
• Figure 4: Logical error rate vs. physical error rate, $[[5,1,3]]$ code, $[[7,1,3]]$ Steane code, $[[9,1,3]]$ Shor code, the $[[13,1,3]]$ surface code, and the $[[41,1,5]]$ surface code, over a depolarizing channel. Comparison between theoretical analysis (curves) and simulation (symbols). The solid curves refer to the asymptotic approximation \ref{['eq:error_probWithBetaApprox']} with the exact $\beta_{t+1}$ from Tab. \ref{['tab:paramBeta']}.
• Figure 5: Logical error rate, $[[23,1,3/5]]$ surface code over symmetric and asymmetric channels. Comparison between: the BD decoding performance \ref{['eq:PeAsym']}; the asymptotic approximation \ref{['eq:betaj_asym']} with the estimated $\hat{\beta}_2$ and $\hat{\beta}_3$ from Tab. \ref{['tab:paramBeta']}, and with their exact values from Tab. \ref{['tab:Err']}; the simulations with a MWPM decoder.

Theorems & Definitions (4)

• Definition 1
• Definition 2: Error class
• Definition 3: Logical operators
• Definition 4: Undetectable errors