Quantum LDPC Codes from Balanced Incomplete Block Designs

Abstract

Abstract-We present a series of structured LDPC codes suitable for use in quantum error correction. Those codes belong to the class of dual-containing Calderbank-Shor-Steane (CSS) codes. The component CSS code is designed using the combinatorial object known as balanced incomplete block design (BIBD) with an even index. The quantum LDPC codes have the rate around 0.9. Several examples of quantum LDPC codes from BIBDs from unity index, extended by addition of an allones column, are introduced as well. To improve the BER performance, we employed the method of removing the cycles of length four in corresponding bipartite graph.

Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength

“…This gives (13) by definition of the hypergraph product: (14) is obtained analogously. Equalities (15) and (16) follow directly.…”

“…In particular, the plain random constructions that work so well in the classical setting are pointless here. There have been a number of attempts at overcoming this difficulty and a variety of methods for constructing quantum LDPC codes have been proposed [33,27,31,10,11,30,19,21,24,15,35,1,2,23,38,25,12,3,4]. However, all of these constructions suffer from disappointingly small minimum distances, namely whenever they have non-vanishing rate and parity-check matrices with bounded row-weight, their minimum distance is either proved to be bounded, or unknown and with little hope for unboundedness.…”

Construction of Quantum LDPC Codes From Classical Row-Circulant QC-LDPCs

“…In this letter, we focus our attention on the construction of QLDPC 42 codes from the family of classical row-circulant quasi-cyclic 43 (QC)-LDPC matrices, which are known to generate efficient 44 high-rate short and moderate-length QC-LDPC codes [5]- [7]. 45 The balanced incomplete block design (BIBD) [6] and the 46 cyclic difference family based LDPC code structures [7] [10], which do not exhibit the unavoidable 61 length-4 cycles in the binary formalism, Djordjevic [11] con-62 ceived the EA counterparts of the BIBD-based designs of 63 [9]. The resultant EA-QLDPC codes required a single pre-64 shared entangled qubit (ebit), which constitutes a valuable 65 resource, because maintaining a noiseless entangled state is 66 not a trivial task.…”

Low-Density Parity-Check Codes and Their Rateless Relatives

“…Recently, Djordjevic also proposed BIBD-based quantum LDPC codes [230] as well as quantum LDPC encoders and decoders for employment in an all-optical implementation [231].…”