Logical Error Rates of XZZX and Rotated Quantum Surface Codes

TL;DR

This paper provides a theoretical analysis of the logical error rates of XZZX and rotated surface codes, revealing how lattice modifications and asymmetries affect their performance in quantum error correction.

Contribution

It introduces formulas for logical error rates of rotated and XZZX surface codes, filling a gap in theoretical understanding and guiding optimal code design.

Findings

Logical error rate approaches 10p^2 for rotated [[9,1,3]] code

Logical error rate approaches 18.3p^2 for [[13,1,3]] surface code

Simultaneous rotation and XZZX modifications can be suboptimal for rectangular lattices.

Abstract

Surface codes are versatile quantum error-correcting codes known for their planar geometry, making them ideal for practical implementations. While the original proposal used Pauli $X$ or Pauli $Z$ operators in a square structure, these codes can be improved by rotating the lattice or incorporating a mix of generators in the XZZX variant. However, a comprehensive theoretical analysis of the logical error rate for these variants has been lacking. To address this gap, we present theoretical formulas based on recent advancements in understanding the weight distribution of stabilizer codes. For example, over an asymmetric channel with asymmetry $A=10$ and a physical error rate $p \to 0$, we observe that the logical error rate asymptotically approaches $p_\mathrm{L} \to 10 p^2$ for the rotated $[[9,1,3]]$ XZZX code and $p_\mathrm{L} \to 18.3 p^2$ for the $[[13,1,3]]$ surface code. Additionally, we observe a particular behavior regarding rectangular lattices in the presence of asymmetric channels. Our findings demonstrate that implementing both rotation and XZZX modifications simultaneously can lead to suboptimal performance. Thus, in scenarios involving a rectangular lattice, it is advisable to avoid using both modifications simultaneously. This research enhances our theoretical understanding of the logical error rates for XZZX and rotated surface codes, providing valuable insights into their performance under different conditions.