Logical Error Rates of XZZX and Rotated Quantum Surface Codes
- URL: http://arxiv.org/abs/2312.17057v1
- Date: Thu, 28 Dec 2023 15:09:48 GMT
- Title: Logical Error Rates of XZZX and Rotated Quantum Surface Codes
- Authors: Diego Forlivesi, Lorenzo Valentini, Marco Chiani
- Abstract summary: We present theoretical formulas based on recent advancements in understanding the weight distribution of stabilizer codes.
We observe that the logical error rate approaches $p_mathrmL to 10 p2$ for the rotated $[9,1,3]]$ XZZX code and $p_mathrmL to 18.3 p2$ for the $[13,1,3]]$ surface code.
Our findings demonstrate that implementing both rotation and XZZX modifications simultaneously can lead to suboptimal performance.
- Score: 9.69910104594168
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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