Topological subsystem codes from graphs and hypergraphs

Topological subsystem codes were proposed by Bombin based on 3-face-colorable cubic graphs. Suchara, Bravyi, and Terhal generalized this construction and proposed a method to construct topological subsystem codes using 3-valent hypergraphs that satisfy certain constraints. Finding such hypergraphs and computing their parameters, however, is a nontrivial task. We propose families of topological subsystem codes that were previously not known. In particular, our constructions give codes which cannot be derived from Bombin's construction. We also study the error recovery schemes for the proposed subsystem codes and give detailed schedules for the syndrome measurement that take advantage of the 2-locality of the gauge group. The study also leads to a new and general construction for color codes.

[1]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[2]  Richard A. Duke.  Types of Cycles in Hypergraphs , 1985 .

[3]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[4]  N. J. A. Sloane,et al.  Quantum Error Correction Via Codes Over GF(4) , 1998, IEEE Trans. Inf. Theory.

[5]  R. Carroll.  North-Holland Mathematics Studies 186 , 2000 .

[6]  A. Levine,et al.  New estimates of the storage permanence and ocean co-benefits of enhanced rock weathering , 2023, PNAS nexus.

[7]  David W. Kribs.  A brief introduction to operator quantum error correction , 2005 .

[8]  D. Poulin.  Stabilizer formalism for operator quantum error correction. , 2005, Physical review letters.

[9]  David Poulin,et al.  Unified and generalized approach to quantum error correction. , 2004, Physical review letters.

[10]  H. Bombin,et al.  Topological quantum distillation. , 2006, Physical review letters.

[11]  David Poulin,et al.  Operator quantum error correction , 2006, Quantum Inf. Comput..

[12]  D. Bacon.  Operator quantum error-correcting subsystems for self-correcting quantum memories , 2005, quant-ph/0506023.

[13]  Panos Aliferis,et al.  Subsystem fault tolerance with the Bacon-Shor code. , 2007, Physical review letters.

[14]  H. Bombin,et al.  Exact topological quantum order in D=3 and beyond : Branyons and brane-net condensates , 2006, cond-mat/0607736.

[15]  Robert Raussendorf,et al.  Fault-tolerant quantum computation with high threshold in two dimensions. , 2007, Physical review letters.

[16]  Pradeep Kiran Sarvepalli,et al.  Clifford Code Constructions of Operator Quantum Error-Correcting Codes , 2006, IEEE Transactions on Information Theory.

[17]  B. Terhal,et al.  A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes , 2008, 0810.1983.

[18]  Austin G. Fowler,et al.  Graphical algorithms and threshold error rates for the 2d color code , 2009, Quantum Inf. Comput..

[19]  H. Bombin,et al.  Topological subsystem codes , 2009, 0908.4246.

[20]  Martin Suchara,et al.  Constructions and noise threshold of topological subsystem codes , 2010, Journal of Physics A: Mathematical and Theoretical.

[21]  Helmut G. Katzgraber,et al.  Optimal error correction in topological subsystem codes , 2012, 1204.1838.