Color codes are a class of topological quantum codes with a high error threshold and a large set of transversal encoded gates and are thus suitable for fault-tolerant quantum computation in two-dimensional architectures. Recently, computationally efficient decoders for the color codes were proposed. We describe an alternate efficient iterative decoder for topological color codes and apply it to the color code on the hexagonal lattice embedded on a torus. In numerical simulations, we find an error threshold of 7.8 $%$ for independent dephasing and spin-flip errors.Color codes are a class of topological quantum codes with a high error threshold and a large set of transversal encoded gates and are thus suitable for fault-tolerant quantum computation in two-dimensional architectures. Recently, computationally efficient decoders for the color codes were proposed. We describe an alternate efficient iterative decoder for topological color codes and apply it to the color code on the hexagonal lattice embedded on a torus. In numerical simulations, we find an error threshold of 7.8$%$ for independent dephasing and spin-flip errors.

arXiv:1111.0831Physical Review AFeb 13, 201251 citations

Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices

Masayuki Ohzeki

Accuracy thresholds of quantum error correcting codes, which exploit topological properties of systems, defined on two different arrangements of qubits are predicted. We study the topological color codes on the hexagonal lattice and on the square-octagonal lattice by the use of mapping into the spin-glass systems. The analysis for the corresponding spin-glass systems consists of the duality, and the gauge symmetry, which has succeeded in deriving locations of special points, which are deeply related with the accuracy thresholds of topological error correcting codes. We predict that the accuracy thresholds for the topological color codes would be 1-pc=0.1096-8 for the hexagonal lattice and 1-pc=0.1092-3 for the square-octagonal lattice, where 1-p denotes the error probability on each qubit. Hence, both of them are expected to be slightly lower than the probability 1-pc=0.110 028 for the quantum Gilbert-Varshamov bound with a zero encoding rate.Accuracy thresholds of quantum error correcting codes, which exploit topological properties of systems, defined on two different arrangements of qubits are predicted. We study the topological color codes on the hexagonal lattice and on the square-octagonal lattice by the use of mapping into the spin-glass systems. The analysis for the corresponding spin-glass systems consists of the duality, and the gauge symmetry, which has succeeded in deriving locations of special points, which are deeply related with the accuracy thresholds of topological error correcting codes. We predict that the accuracy thresholds for the topological color codes would be 1-pc=0.1096-8 for the hexagonal lattice and 1-pc=0.1092-3 for the square-octagonal lattice, where 1-p denotes the error probability on each qubit. Hence, both of them are expected to be slightly lower than the probability 1-pc=0.110 028 for the quantum Gilbert-Varshamov bound with a zero encoding rate.

arXiv:0903.2102Physical Review EJul 30, 200940 citations

Topological Stabilizer Codes

Keisuke Fujii

SpringerBriefs in mathematical physicsJan 1, 20150 citations

The boundaries and topological defects of the color code

Markus S. Kesselring, Fernando Pastawski, Jens Eisert +1 more

Bulletin of the American Physical SocietyMar 7, 20190 citations

Twins Percolation for Qubit Losses in Topological Color Codes

Davide Vodola, David Amaro, Miguel Angel Martin-Delgado +1 more

In this Letter, we establish and explore a new connection between quantum information theory and classical statistical mechanics by studying the problem of qubit losses in 2D topological color codes. We introduce a protocol to cope with qubit losses, which is based on the identification and removal of a twin qubit from the code, and which guarantees the recovery of a valid three-colorable and trivalent reconstructed color code. Moreover, we show that determining the corresponding qubit loss error threshold is equivalent to a new generalized classical percolation problem. We numerically compute the associated qubit loss thresholds for two families of 2D color code and find that with p=0.461±0.005 these are close to satisfying the fundamental limit of 50% as imposed by the no-cloning theorem. Our findings reveal a new connection between topological color codes and percolation theory, show high robustness of color codes against qubit loss, and are directly relevant for implementations of topological quantum error correction in various physical platforms.In this Letter, we establish and explore a new connection between quantum information theory and classical statistical mechanics by studying the problem of qubit losses in 2D topological color codes. We introduce a protocol to cope with qubit losses, which is based on the identification and removal of a twin qubit from the code, and which guarantees the recovery of a valid three-colorable and trivalent reconstructed color code. Moreover, we show that determining the corresponding qubit loss error threshold is equivalent to a new generalized classical percolation problem. We numerically compute the associated qubit loss thresholds for two families of 2D color code and find that with p=0.461±0.005 these are close to satisfying the fundamental limit of 50% as imposed by the no-cloning theorem. Our findings reveal a new connection between topological color codes and percolation theory, show high robustness of color codes against qubit loss, and are directly relevant for implementations of topological quantum error correction in various physical platforms.

arXiv:1802.04545Physical Review LettersAug 6, 201818 citations

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for independent dephasing and spin-flip errors.