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Quantum Quasi-Cyclic Low-Density Parity-Check Codes

Min-Hsiu Hsieh, Todd A. Brun, Igor Devetak

Published Mar 2, 2008• 8 citations
OpenAlex

Abstract

We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Shor-Steane (CSS) construction do not need to satisfy the dual-containing property as long as pre-shared entanglement is available to both sender and receiver. We can use this to avoid the many 4-cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product algorithm (SPA). It is well known that in the SPA, cycles of length 4 make successive decoding iterations highly correlated and hence limit the decoding performance. We show the principle of constructing quantum QC-LDPC codes which require only small amounts of initial shared entanglement.

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International Conference on Signal Processing•Oct 1, 2012•4 citations
We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Shor-Steane (CSS) construction do not need to satisfy the dual-containing property as long as pre-shared entanglement is available to both sender and receiver. We can use this to avoid the many 4-cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product algorithm (SPA). It is well known that in the SPA, cycles of length 4 make successive decoding iterations highly correlated and hence limit the decoding performance. We show the principle of constructing quantum QC-LDPC codes which require only small amounts of initial shared entanglement.
We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasicyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Skor-Steane construction do not need to satisfy the dual-containing property as long as preshared entanglement is available to both sender and receiver. We can use this to avoid the many four cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product algorithm (SPA). It is well known that in the SPA, cycles of length 4 make successive decoding iterations highly correlated and hence limit the decoding performance. We show the principle of constructing quantum QC-LDPC codes which require only small amounts of initial shared entanglement.
Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank-Shor-Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.
We provide code constructions along with code properties for a few families of entanglement-assisted (EA) Calderbank-Shor-Steane (CSS) codes derived from quasi-cyclic (QC) low-density parity check (LDPC) codes. These codes require only a minimum number of Bell pairs to be shared between the transmitter and the receiver. The Tanner graph of the proposed EA QC LDPC code has girth>4> 4>4. We evaluate the performance of these codes using both random and Markovian noise models to assess random and burst error performance using a modified-version of the min-sum algorithm.
Quantum quasi-cyclic (QC) and quantum low-density parity-check (LDPC) codes have received significant attention due to their algebraic regularity and performance under low-complexity decoding. In this work, we explore a class of structured quantum codes based on reproducible matrices, which generalize known families such as cyclic, QC, and dyadic codes. Focusing on sparse quasi-dyadic (QD) structures, we investigate how their symmetries influence Tanner graph cycles. Moreover, we construct quantum LDPC codes within the Calderbank-Shor-Steane (CSS) framework, analyzing dual-containing configurations, and assess their error rates.
Quantum error correcting codes are essential to achieve fault-tolerant quantum computation. This work introduces two constructions of high-rate, dual-containing (DC) Calderbank--Shor--Steane low-density parity-check (LDPC) codes based on quasi-dyadic matrices. We characterize the automorphism group of such codes, investigate their minimum distance behavior, and provide several theoretical results on their cycle properties. Monte Carlo simulations under depolarizing and phenomenological noise show better finite-length logical error rates than the considered DC benchmark codes and competitive performance against several state-of-the-art quantum LDPC code families. Finally, we employ an automorphism-ensemble belief propagation decoder to improve their decoding performance.
As the best error correction codes in classical communications, low density parity check codes (LDPC) have found its usage in quantum error correction. In this paper, we investigate a family of entanglement-assisted quantum LDPC codes based on cyclic difference set. It is shown that these codes only need one pre-shared entangled qubit between both sender and receiver, without the necessity to satisfy the dual-containing property in the generalized Calderban-Shor-Stean (CSS) construction. The simulation results show that the quantum LDPC codes have better performance than that quantum LDPC codes on cyclic difference set with the dual-containing quantum CSS construction.