3D subsystem color code[1]
Alternative names: 3D gauge color code.
Description
A subsystem version of the 3D color code defined on a 3-colex.
In the tetrahedral subsystem family introduced in [1], qubits live on tetrahedra, boundary triangles, boundary edges, and boundary vertices of a colored tetrahedron, and the code encodes one logical qubit. Gauge generators can be chosen to have weight four or six, while the corresponding stabilizer generators can be much larger.
Transversal Gates
For the \((1,1)\) member, \(CNOT\) and Hadamard are transversal; gauge fixing to the \((1,2)\) code enables a transversal \(R_3\) gate, yielding a universal gate set without encoded ancillas [1].Fault Tolerance
In the tetrahedral subsystem family, syndrome extraction can use 4- or 6-qubit gauge checks instead of directly measuring larger stabilizers, and gauge fixing uses only local quantum operations plus classical processing [1].Threshold
Phenomenological noise: \(0.31\%\) under clustering decoder [2].Cousins
- 3D color code— On a fixed 3D lattice, the 3D subsystem color code is gauge-related to the 3D color code; switching between the \((1,1)\) and \((1,2)\) members yields transversal \(CNOT\), \(H\), and \(R_3\) gates [1].
- Single-shot code— The 3D subsystem color code defined on the cube-truncated rhombic dodecahedral honeycomb, i.e., a tessellation 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].
- 3D surface code— The 3D subsystem color code can be ungauged [5,7–15] to obtain six copies of \(\mathbb{Z}_2\) gauge theory with one-form symmetries [5].
- Symmetry-protected topological (SPT) code— Ungauging [5,7–15] different stabilizer Hamiltonians of the 3D subsystem color code yields distinct SPT phases; in particular, one ungauges to three decoupled copies of the RBH model [5].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code— Different stabilizer Hamiltonians of the 3D subsystem color code correspond to distinct SPT phases; one ungauges to three decoupled copies of 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].
Primary Hierarchy
Parents
3D subsystem color code
References
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- K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
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- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
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- D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
Page edit log
- Victor V. Albert (2024-07-11) — most recent
Cite as:
“3D subsystem color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/3d_subsystem_color