A Pair Measurement Surface Code on Pentagons
Craig Gidney
Abstract
In this paper, I present a way to compile the surface code into two-body parity measurements ("pair measurements"), where the pair measurements run along the edges of a Cairo pentagonal tiling. The resulting circuit improves on prior work by Chao et al. by using fewer pair measurements per four-body stabilizer measurement (5 instead of 6) and fewer time steps per round of stabilizer measurement (6 instead of 10). Using Monte Carlo sampling, I show that these improvements increase the threshold of the surface code when compiling into pair measurements from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2248;</mml:mo><mml:mn>0.2</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2248;</mml:mo><mml:mn>0.4</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math>, and also that they improve the teraquop footprint at a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.1</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math> physical gate error rate from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2248;</mml:mo><mml:mn>6000</mml:mn></mml:math> qubits to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2248;</mml:mo><mml:mn>3000</mml:mn></mml:math> qubits. However, I also show that the teraquop footprint of Chao et al's construction improves more quickly than mine as physical error rate decreases, and is likely better below a physical gate error rate of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2248;</mml:mo><mml:mn>0.03</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math> (due to bidirectional hook errors in my construction). I also compare to the planar honeycomb code, showing that although this work does noticeably reduce the gap between the surface code and the honeycomb code (when compiling into pair measurements), the honeycomb code is still more efficient (threshold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2248;</mml:mo><mml:mn>0.8</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math>, teraquop footprint at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.1</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2248;</mml:mo><mml:mn>1000</mml:mn></mml:math>).