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

In the absence of fault tolerant quantum error correction for analog, Hamiltonian quantum computation, error suppression via energy penalties is an effective alternative. We construct families of distance-$2$ stabilizer subsystem codes we call ``trapezoid codes'', that are tailored for energy-penalty schemes. We identify a family of codes achieving the maximum code rate, and by slightly relaxing this constraint, uncover a broader range of codes with enhanced physical locality, thus increasing their practical applicability. Additionally, we provide an algorithm to map the required qubit connectivity graph into graphs compatible with the locality constraints of quantum hardware. Finally, we provide a systematic framework to evaluate the performance of these codes in terms of code rate, physical locality, graph properties, and penalty gap, enabling an informed selection of error-suppression codes for specific quantum computing applications. We identify the $[[4k+2,2k,g,2]]$ family of subsystem codes as optimal in terms of code rate and penalty gap scaling.