Planar quantum low-density parity-check codes with open boundaries

We construct high-performance planar quantum low-density parity-check (qLDPC) codes with open boundaries, demonstrating substantially improved resource efficiency compared to the surface code. We present planar code families with logical dimensions ranging from $k=6$ to $k=13$ (e.g., $[[79, 6, 6]]$, $[[107, 7, 7]]$, $[[173, 8, 9]]$, $[[268, 8, 12]]$, $[[405, 9, 15]]$, $[[374, 10, 13]]$, $[[409, 11, 13]]$, $[[386, 12, 12]]$, $[[362, 13, 11]]$), all using local stabilizers of weight 6 or lower. These codes achieve an efficiency metric ($kd^2/n$) that is an order of magnitude greater than that of the surface code. They can be interpreted as planar bivariate bicycle codes, adapted from the original design based on a torus that is challenging to implement physically. Our construction method, which combines boundary anyon condensation with a novel "lattice grafting" optimization, circumvents this difficulty and produces codes featuring only local low-weight stabilizers suitable for 2D planar hardware architectures. Furthermore, we observe fractal logical operators in the form of Sierpinski triangles, with the code distances scaling proportionally to the area of the truncated fractal in finite systems. We anticipate that our codes and construction methods offer a promising pathway toward realizing near-term fault-tolerant quantum computers.