Families of d=2 2D subsystem stabilizer codes for universal adiabatic quantum computation with two-body interactions

Bravyi's A matrix offers an approach to devising quantum error correction codes (QECC) characterized by geometric constraints. Since two-body interactions are sufficient for universal adiabatic quantum computation (AQC), we focus on the quantum error detection code (QEDC) with d=2. We discovered a family of codes satisfying the maximum code rate, and by slightly relaxing the code rate, we uncovered an extended spectrum of codes within this framework. These codes present enhanced geometric locality, which amplifies their practical utility. Furthermore, we also map the requisite connectivity to alternative configurations so that the total Manhattan distance is minimized, providing valuable insights into hardware design. Lastly, we give a systematic framework for the assessment of codes within the context of AQC in terms of code rate, physical and geometrical locality, graph complexity, and Manhattan distances on the graph. This facilitates informed decision-making in code selection for specific quantum computing applications.