Quantum Tanner codes

TL;DR

This paper constructs quantum error-correcting codes using Tanner codes on a special complex, achieving improved minimum distance and maintaining local testability, advancing quantum LDPC code design.

Contribution

It introduces a novel quantum Tanner code based on the left-right Cayley complex, improving minimum distance estimates and preserving local testability.

Findings

Quantum Tanner codes with linearly growing minimum distance.

Improved estimates for quantum code minimum distance.

Preservation of local testability in the quantum code.

Abstract

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 al. 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 al. 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 al. code.