3D subsystem color code[1]

Cousins

• 3D color code
• Single-shot code— The 3D subsystem color code defined on the cube-truncated rhombic dodecahedral honeycomb, i.e., a tesselation of cubes and chamfered cubes (a.k.a. tetratruncated rhombic dodecahedra) [2; Fig. 1], is a single-shot code [2,3].
• Symmetry-protected self-correcting quantum code— A particular gauge-fixed version of a subsystem code on a 3D lattice yields a self-correcting memory protected by one-form symmetries [5][4; Sec. IV D]. The symmetric energy barrier grows linearly with the length of a side of the lattice. When the system is coupled locally to a thermal bath respecting the symmetry and below a critical temperature, the memory time grows exponentially with the side length. The subsystem color code is not a self-correcting quantum memory if symmetry protection is removed [6].
• Two-gauge theory code— The 3D subsystem color code can be ungauged [7–9,9] to obtain six copies of \(\mathbb{Z}_2\) gauge theory with one-form symmetries [5].
• Symmetry-protected topological (SPT) code— Different stabilizer Hamiltonians of the 3D subsystem color code correspond to different SPTs, one of which describes the RBH model [5].
• Raussendorf-Bravyi-Harrington (RBH) cluster-state code— Different stabilizer Hamiltonians of the 3D subsystem color code correspond to different SPTs, one of which describes the RBH model [5]. The RBH code for a certain boundary Hamiltonian is dual to the 3D subsystem color code [4; Sec. IV.C.1].