Quantum to classical mapping of the two-dimensional toric code in an external field
Kitaev's toric code Hamiltonian in dimension D=2 has been extensively studied for its topological properties, including its quantum error correction capabilities. While the Hamiltonian is quantum, it lies within the class of models that admits a D+1 dimensional classical representation. In thes...
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| Format: | Article |
| Language: | English |
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SciPost
2022-07-01
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| Series: | SciPost Physics Lecture Notes |
| Online Access: | https://scipost.org/SciPostPhysLectNotes.57 |
| _version_ | 1828403853895139328 |
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| author | Sydney R. Timmerman, Zvonimir Z. Bandic, Roger G. Melko |
| author_facet | Sydney R. Timmerman, Zvonimir Z. Bandic, Roger G. Melko |
| author_sort | Sydney R. Timmerman, Zvonimir Z. Bandic, Roger G. Melko |
| collection | DOAJ |
| description | Kitaev's toric code Hamiltonian in dimension D=2 has been extensively studied for its topological properties, including its quantum error correction capabilities. While the Hamiltonian is quantum, it lies within the class of models that admits a D+1 dimensional classical representation. In these notes, we provide details of a Suzuki-Trotter expansion of the partition function of the toric code Hamiltonian in the presence of an external magnetic field. By coupling additional degrees of freedom in the form of a matter field that can subsequently be gauged away, we explicitly derive a classical Hamiltonian on a cubic lattice which takes the form of a non-isotropic 3D Ising gauge theory. |
| first_indexed | 2024-12-10T10:25:56Z |
| format | Article |
| id | doaj.art-a1175cd5bc8449a383986af8a6100156 |
| institution | Directory Open Access Journal |
| issn | 2590-1990 |
| language | English |
| last_indexed | 2024-12-10T10:25:56Z |
| publishDate | 2022-07-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics Lecture Notes |
| spelling | doaj.art-a1175cd5bc8449a383986af8a61001562022-12-22T01:52:44ZengSciPostSciPost Physics Lecture Notes2590-19902022-07-015710.21468/SciPostPhysLectNotes.57Quantum to classical mapping of the two-dimensional toric code in an external fieldSydney R. Timmerman, Zvonimir Z. Bandic, Roger G. MelkoKitaev's toric code Hamiltonian in dimension D=2 has been extensively studied for its topological properties, including its quantum error correction capabilities. While the Hamiltonian is quantum, it lies within the class of models that admits a D+1 dimensional classical representation. In these notes, we provide details of a Suzuki-Trotter expansion of the partition function of the toric code Hamiltonian in the presence of an external magnetic field. By coupling additional degrees of freedom in the form of a matter field that can subsequently be gauged away, we explicitly derive a classical Hamiltonian on a cubic lattice which takes the form of a non-isotropic 3D Ising gauge theory.https://scipost.org/SciPostPhysLectNotes.57 |
| spellingShingle | Sydney R. Timmerman, Zvonimir Z. Bandic, Roger G. Melko Quantum to classical mapping of the two-dimensional toric code in an external field SciPost Physics Lecture Notes |
| title | Quantum to classical mapping of the two-dimensional toric code in an external field |
| title_full | Quantum to classical mapping of the two-dimensional toric code in an external field |
| title_fullStr | Quantum to classical mapping of the two-dimensional toric code in an external field |
| title_full_unstemmed | Quantum to classical mapping of the two-dimensional toric code in an external field |
| title_short | Quantum to classical mapping of the two-dimensional toric code in an external field |
| title_sort | quantum to classical mapping of the two dimensional toric code in an external field |
| url | https://scipost.org/SciPostPhysLectNotes.57 |
| work_keys_str_mv | AT sydneyrtimmermanzvonimirzbandicrogergmelko quantumtoclassicalmappingofthetwodimensionaltoriccodeinanexternalfield |