Quantum Tanner codes

Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes. In this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a c 3 locally testable code by Dinur et at. Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur et at. code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et at. code.

Authors

• Anthony Leverrier
• Gilles Zémor

Keywords

• Heart
• Computer science
• Codes
• Quantum computing
• Quantum mechanics
• Parity check codes
• Graph theory
• Minimum Distance
• Presence Of Complex
• Low-density Parity-check Codes
• Quantum Error Correction
• Low-density Parity-check
• Short Codes
• Cardinality
• Types Of Errors
• General Setting
• Sparse Matrix
• Constant Distance
• Multiple Edges
• Code Length
• Number Of Squares
• Linear Code
• Weight Error
• Regular Graphs
• Local View
• Dual Coding
• Reed-Solomon Codes
• Parity-check Matrix
• Ambient Space
• Quantum Case
• Code Construction
• Large Distances
• Balance Of Production
• Family Of Codes
• Quantum error correcting code