Logical Error Rates of XZZX and Rotated Quantum Surface Codes
Diego Forlivesi, Lorenzo Valentini, Marco Chiani
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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> or Pauli <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> = 10 and a physical error rate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> → 0, we observe that the logical error rate asymptotically approaches <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> → 10 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> for the rotated [[9, 1, 3]] XZZX code and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> → 18.3 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> 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.