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Error threshold in topological quantum-computing models with color codes

Helmut G. Katzgraber, Héctor Bombín, Miguel Angel Martin-Delgado

Bulletin of the American Physical SocietyMar 19, 20090 citations

Error Threshold for the 4.6.12 Topological Color Code

Colin J. Trout

Bulletin of the American Physical SocietyMar 15, 20160 citations

Topological color codes on Union Jack lattices: a stable implementation of the whole Clifford group

Helmut G. Katzgraber, Héctor Bombín, Ruben S. Andrist +1 more

We study the error threshold of topological color codes on Union Jack lattices that allow for the full implementation of the whole Clifford group of quantum gates. After mapping the error-correction process onto a statistical mechanical random three-body Ising model on a Union Jack lattice, we compute its phase diagram in the temperature--disorder plane using Monte Carlo simulations. Surprisingly, topological color codes on Union Jack lattices have a similar error stability to color codes on triangular lattices, as well as to the Kitaev toric code. The enhanced computational capabilities of the topological color codes on Union Jack lattices with respect to triangular lattices and the toric code combined with the inherent robustness of this implementation show good prospects for future stable quantum computer implementations.We study the error threshold of topological color codes on Union Jack lattices that allow for the full implementation of the whole Clifford group of quantum gates. After mapping the error-correction process onto a statistical mechanical random three-body Ising model on a Union Jack lattice, we compute its phase diagram in the temperature--disorder plane using Monte Carlo simulations. Surprisingly, topological color codes on Union Jack lattices have a similar error stability to color codes on triangular lattices, as well as to the Kitaev toric code. The enhanced computational capabilities of the topological color codes on Union Jack lattices with respect to triangular lattices and the toric code combined with the inherent robustness of this implementation show good prospects for future stable quantum computer implementations.

arXiv:0910.0573Physical Review AJan 22, 201031 citations

Topological stabilizer code on a honeycomb lattice

Basudha Srivastava

Jan 1, 20220 citations

Statistical mechanical models and topological color codes

Héctor Bombín, Miguel Angel Martin-Delgado

We find that the overlap of a topological quantum color-code state, representing a quantum memory, with a factorized state of qubits can be written as the partition function of a three-body classical Ising model on triangular or Union Jack lattices. This mapping allows us to test that different computational capabilities of color codes correspond to qualitatively different universality classes of their associated classical spin models. By generalizing these statistical mechanical models for arbitrary inhomogeneous and complex couplings, it is possible to study a measurement-based quantum computation with a color-code state and we find that their classical simulability remains an open problem. We complement the measurement-based computation with the construction of a cluster state that yields the topological color code and this also gives the possibility to represent statistical models with external magnetic fields.We find that the overlap of a topological quantum color-code state, representing a quantum memory, with a factorized state of qubits can be written as the partition function of a three-body classical Ising model on triangular or Union Jack lattices. This mapping allows us to test that different computational capabilities of color codes correspond to qualitatively different universality classes of their associated classical spin models. By generalizing these statistical mechanical models for arbitrary inhomogeneous and complex couplings, it is possible to study a measurement-based quantum computation with a color-code state and we find that their classical simulability remains an open problem. We complement the measurement-based computation with the construction of a cluster state that yields the topological color code and this also gives the possibility to represent statistical models with external magnetic fields.

arXiv:0711.0468Physical Review AApr 24, 200839 citations

Error-threshold for topological subsystem quantum error-correcting codes

Ruben S. Andrist, Héctor Bombín, Miguel Angel Martin-Delgado +1 more

Bulletin of the American Physical SocietyFeb 27, 20120 citations

Error threshold for topological color codes on Union Jack lattices

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