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Journal of Communications and Networks

Novel class of entanglement-assisted quantum codes with minimal ebits

Cao Dong, Yaoliang Song

Published Apr 1, 2013• 3 citations
Publisher

Abstract

Quantum low-density parity-check (LDPC) codes based on the Calderbank-Shor-Steane construction have low encoding and decoding complexity. The sum-product algorithm (SPA) can be used to decode quantum LDPC codes; however, the decoding performance may be significantly decreased by the many four-cycles required by this type of quantum codes. All four-cycles can be eliminated using the entanglement-assisted formalism with maximally entangled states (ebits). The proposed entanglement-assisted quantum error-correcting code based on Euclidean geometry outperform differently structured quantum codes. However, the large number of ebits required to construct the entanglement-assisted formalism is a substantial obstacle to practical application. In this paper, we propose a novel class of entanglement-assisted quantum LDPC codes constructed using classical Euclidean geometry LDPC codes. Notably, the new codes require one copy of the ebit. Furthermore, we propose a construction scheme for a corresponding zigzag matrix and show that the algebraic structure of the codes could easily be expanded. A large class of quantum codes with various code lengths and code rates can be constructed. Our methods significantly improve the possibility of practical implementation of quantum error-correcting codes. Simulation results show that the entanglement-assisted quantum LDPC codes described in this study perform very well over a depolarizing channel with iterative decoding based on the SPA and that these codes outperform other quantum codes based on Euclidean geometries.

DOI
10.1109/jcn.2013.000035

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Quantum low-density parity-check (LDPC) codes based on the Calderbank-Shor-Steane construction have low encoding and decoding complexity. The sum-product algorithm (SPA) can be used to decode quantum LDPC codes; however, the decoding performance may be significantly decreased by the many four-cycles required by this type of quantum codes. All four-cycles can be eliminated using the entanglement-assisted formalism with maximally entangled states (ebits). The proposed entanglement-assisted quantum error-correcting code based on Euclidean geometry outperform differently structured quantum codes. However, the large number of ebits required to construct the entanglement-assisted formalism is a substantial obstacle to practical application. In this paper, we propose a novel class of entanglement-assisted quantum LDPC codes constructed using classical Euclidean geometry LDPC codes. Notably, the new codes require one copy of the ebit. Furthermore, we propose a construction scheme for a corresponding zigzag matrix and show that the algebraic structure of the codes could easily be expanded. A large class of quantum codes with various code lengths and code rates can be constructed. Our methods significantly improve the possibility of practical implementation of quantum error-correcting codes. Simulation results show that the entanglement-assisted quantum LDPC codes described in this study perform very well over a depolarizing channel with iterative decoding based on the SPA and that these codes outperform other quantum codes based on Euclidean geometries.
As in classical coding theory, quantum analogs of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the existence of quantum LDPC codes which are obtainable from pairs of identical LDPC codes and consume only one ebit, and studies the spectrum of attainable code parameters.
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.
A new class of quantum low-density parity-check (LDPC) codes whose parity-check matrices are dual-containing matrices constructed based on lines of Euclidean geometries (EGs) is presented. The parity-check matrices of our quantum codes contain one and only one 4-cycle in every two rows and have better distance properties. However, the classical parity-check matrix constructed from EGs does not satisfy the condition of dual-containing. In some parameter conditions, parts of the rows in the matrix maybe have not any nonzero element in common. Notably, we propose four families of fascinating structure according to changes in all the parameters, and the parity-check matrices are adopted to satisfy the requirement of dual-containing. Series of matrix properties are proved. Construction methods of the parity-check matrices with dual-containing property are given. The simulation results show that the quantum LDPC codes constructed by this method perform very well over the depolarizing channel when decoded with iterative decoding based on the sum-product algorithm. Also, the quantum codes constructed in this paper outperform other quantum codes based on EGs.
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 construct several families of entanglement-assisted quasi-cyclic quantum LDPC (EA-QC-QLDPC) codes via structured tilings of permutation matrices. The entanglement-unassisted portion of the joint Tanner graph of the proposed EA-QC-QLDPC code derived from two distinct classical QC-LDPC codes is free of 4-cycles. Notably, one of the proposed families constructed from two distinct classical codes requires only a s i n g l e shared Bell pair between the quantum transmitter and receiver, highlighting its resource efficiency. We also analytically determine the exact code rates for some of the proposed constructions. Furthermore, two of the proposed families of EA-QC-QLDPC codes are derived from a single classical code whose Tanner graphs possess girth greater than six, further enhancing their error-correcting performance.We also propose an encoding scheme with improved complexity by exploiting the proposed code structure. The performance of the proposed codes is assessed under both random and burst error models under the depolarizing and Markovian noise actions. Simulation results reveal nearly one order of improvement in error-correction performance with the quaternary block-layered normalized min-sum (QBLNMS) decoder compared to the layered binary sum-product decoder over both depolarizing and Markovian channels. Using the QBLNMS decoder over a quaternary alphabet, we demonstrate that correlated Pauli errors can be effectively handled within the decoding framework.Furthermore, under the QBLNMS decoding, the proposed codes achieve s i g n i f i c a n t performance improvements compared to prior works and can effectively handle both random and burst errors. The code constructions are scalable across various coding rates and quantum payloads, crucial for practical quantum communication and computing systems.
Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.