Tile Codes: High-Efficiency Quantum Codes on a Lattice with Boundary

We introduce , a simple yet powerful way of constructing quantum codes that are local on a planar 2D lattice. Tile codes generalize the usual surface code by allowing for a bit more flexibility in terms of locality and stabilizer weight. Our construction does not compromise on the fact that the codes are local on a lattice with open boundary conditions. Despite its simplicity, we use our construction to find codes with parameters <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo stretchy="false">[</a:mo><a:mo stretchy="false">[</a:mo><a:mn>288</a:mn><a:mo>,</a:mo><a:mn>8</a:mn><a:mo>,</a:mo><a:mn>12</a:mn><a:mo stretchy="false">]</a:mo><a:mo stretchy="false">]</a:mo></a:math> using weight-6 stabilizers and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo stretchy="false">[</g:mo><g:mo stretchy="false">[</g:mo><g:mn>288</g:mn><g:mo>,</g:mo><g:mn>8</g:mn><g:mo>,</g:mo><g:mn>14</g:mn><g:mo stretchy="false">]</g:mo><g:mo stretchy="false">]</g:mo></g:math> using weight-8 stabilizers, outperforming all previously known constructions in this direction. Allowing for a slightly higher nonlocality, we find a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mo stretchy="false">[</m:mo><m:mo stretchy="false">[</m:mo><m:mn>512</m:mn><m:mo>,</m:mo><m:mn>18</m:mn><m:mo>,</m:mo><m:mn>19</m:mn><m:mo stretchy="false">]</m:mo><m:mo stretchy="false">]</m:mo></m:math> code using weight-8 stabilizers, which outperforms the rotated surface code by a factor of more than 12. Our approach provides a unified framework for understanding the structure of codes that are local on a 2D planar lattice and offers a systematic way to explore the space of possible code parameters. In particular, due to its simplicity, the construction naturally accommodates various types of boundary conditions and stabilizer configurations, making it a versatile tool for quantum error-correction code design.