A Pair Measurement Surface Code on Pentagons

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 $\approx 0.2\%$ to $\approx 0.4\%$, and also that they improve the teraquop footprint at a $0.1\%$ physical gate error rate from $\approx6000$ qubits to $\approx3000$ 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 $\approx 0.03\%$ (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 $\approx 0.8\%$, teraquop footprint at $0.1\%$ of $\approx 1000$).