Topological color codes and two-body quantum lattice Hamiltonians
Topological color codes are among the stabilizer codes with remarkable properties from the quantum information perspective. In this paper, we construct a lattice, the so-called ruby lattice, with coordination number 4 governed by a two-body Hamiltonian. In a particular regime of coupling constants,...
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| Format: | Article |
| Language: | en_US |
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Institute of Physics Publishing
2012
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| Online Access: | http://hdl.handle.net/1721.1/70133 |
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| author | Kargarian, M. Bombin, Hector Martin-Delgado, M. A. |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Kargarian, M. Bombin, Hector Martin-Delgado, M. A. |
| author_sort | Kargarian, M. |
| collection | MIT |
| description | Topological color codes are among the stabilizer codes with remarkable properties from the quantum information perspective. In this paper, we construct a lattice, the so-called ruby lattice, with coordination number 4 governed by a two-body Hamiltonian. In a particular regime of coupling constants, in a strong coupling limit, degenerate perturbation theory implies that the low-energy spectrum of the model can be described by a many-body effective Hamiltonian, which encodes the color code as its ground state subspace. Ground state subspace corresponds to a vortex-free sector. The gauge symmetry Z2×Z2 of the color code could already be realized by identifying three distinct plaquette operators on the ruby lattice. All plaquette operators commute with each other and with the Hamiltonian being integrals of motion. Plaquettes are extended to closed strings or string-net structures. Non-contractible closed strings winding the space commute with Hamiltonian but not always with each other. This gives rise to exact topological degeneracy of the model. A connection to 2-colexes can be established via the coloring of the strings. We discuss it at the non-perturbative level. The particular structure of the two-body Hamiltonian provides a fruitful interpretation in terms of mapping onto bosons coupled to effective spins. We show that high-energy excitations of the model have fermionic statistics. They form three families of high-energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. The emergence of invisible charges is related to the string-net structure of the model. The emerging fermions are coupled to nontrivial gauge fields. We show that for particular 2-colexes, the fermions can see the background fluxes in the ground state. Also, we use the Jordan–Wigner transformation in order to test the integrability of the model via introducing Majorana fermions. The four-valent structure of the lattice prevents the fermionized Hamiltonian from being reduced to a quadratic form owing to interacting gauge fields. We also propose another construction for the two-body Hamiltonian based on the connection between color codes and cluster states. The corresponding two-body Hamiltonian encodes a cluster state defined on a bipartite lattice as its low-energy spectrum, and subsequent selective measurements give rise to the color code model. We discuss this latter approach along with the construction based on the ruby lattice. |
| first_indexed | 2024-09-23T13:12:30Z |
| format | Article |
| id | mit-1721.1/70133 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T13:12:30Z |
| publishDate | 2012 |
| publisher | Institute of Physics Publishing |
| record_format | dspace |
| spelling | mit-1721.1/701332022-09-28T12:37:52Z Topological color codes and two-body quantum lattice Hamiltonians Kargarian, M. Bombin, Hector Martin-Delgado, M. A. Massachusetts Institute of Technology. Department of Physics Bombin, Hector Bombin, Hector Topological color codes are among the stabilizer codes with remarkable properties from the quantum information perspective. In this paper, we construct a lattice, the so-called ruby lattice, with coordination number 4 governed by a two-body Hamiltonian. In a particular regime of coupling constants, in a strong coupling limit, degenerate perturbation theory implies that the low-energy spectrum of the model can be described by a many-body effective Hamiltonian, which encodes the color code as its ground state subspace. Ground state subspace corresponds to a vortex-free sector. The gauge symmetry Z2×Z2 of the color code could already be realized by identifying three distinct plaquette operators on the ruby lattice. All plaquette operators commute with each other and with the Hamiltonian being integrals of motion. Plaquettes are extended to closed strings or string-net structures. Non-contractible closed strings winding the space commute with Hamiltonian but not always with each other. This gives rise to exact topological degeneracy of the model. A connection to 2-colexes can be established via the coloring of the strings. We discuss it at the non-perturbative level. The particular structure of the two-body Hamiltonian provides a fruitful interpretation in terms of mapping onto bosons coupled to effective spins. We show that high-energy excitations of the model have fermionic statistics. They form three families of high-energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. The emergence of invisible charges is related to the string-net structure of the model. The emerging fermions are coupled to nontrivial gauge fields. We show that for particular 2-colexes, the fermions can see the background fluxes in the ground state. Also, we use the Jordan–Wigner transformation in order to test the integrability of the model via introducing Majorana fermions. The four-valent structure of the lattice prevents the fermionized Hamiltonian from being reduced to a quadratic form owing to interacting gauge fields. We also propose another construction for the two-body Hamiltonian based on the connection between color codes and cluster states. The corresponding two-body Hamiltonian encodes a cluster state defined on a bipartite lattice as its low-energy spectrum, and subsequent selective measurements give rise to the color code model. We discuss this latter approach along with the construction based on the ruby lattice. 2012-04-25T19:02:04Z 2012-04-25T19:02:04Z 2010-02 2009-06 Article http://purl.org/eprint/type/JournalArticle 1367-2630 http://hdl.handle.net/1721.1/70133 Kargarian, M, H Bombin, and M A Martin-Delgado. “Topological Color Codes and Two-body Quantum Lattice Hamiltonians.” New Journal of Physics 12.2 (2010): 025018. Web. en_US http://dx.doi.org/10.1088/1367-2630/12/2/025018 New Journal of Physics Creative Commons Attribution 3.0 http://creativecommons.org/licenses/by/3.0/ application/pdf Institute of Physics Publishing New Journal of Physics |
| spellingShingle | Kargarian, M. Bombin, Hector Martin-Delgado, M. A. Topological color codes and two-body quantum lattice Hamiltonians |
| title | Topological color codes and two-body quantum lattice Hamiltonians |
| title_full | Topological color codes and two-body quantum lattice Hamiltonians |
| title_fullStr | Topological color codes and two-body quantum lattice Hamiltonians |
| title_full_unstemmed | Topological color codes and two-body quantum lattice Hamiltonians |
| title_short | Topological color codes and two-body quantum lattice Hamiltonians |
| title_sort | topological color codes and two body quantum lattice hamiltonians |
| url | http://hdl.handle.net/1721.1/70133 |
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