Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities
CoRR(2023)
摘要
Quantum error correcting codes are of primary interest for the evolution
towards quantum computing and quantum Internet. We analyze the performance of
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. Finally, we introduce a novel
approach that combines the knowledge of WE with a logical operator analysis.
This method allows the derivation of the exact asymptotic performance for short
codes. For example, on a depolarizing channel with physical error rate ρ→ 0 it is found that the logical error rate ρ_L is
asymptotically ρ_L≈ 16 ρ^2 for the [[9,1,3]] Shor
code, ρ_L≈ 16.3 ρ^2 for the [[7,1,3]] Steane code,
ρ_L≈ 18.7 ρ^2 for the [[13,1,3]] surface code, and
ρ_L≈ 149.3 ρ^3 for the [[41,1,5]] surface code. For
larger codes our bound provides ρ_L≈ 1215 ρ^4 and
ρ_L≈ 663 ρ^5 for the [[85,1,7]] and the
[[181,1,10]] surface codes, respectively.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要