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International Conference on Signal Processing

A construction of entanglement-assisted quantum LDPC codes from the cyclic difference set

Ying Liu, Yanyan Wang, Shengmei Zhao, Baoyu Zheng

Published Oct 1, 2012• 4 citations
Publisher

Abstract

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.

DOI
10.1109/icosp.2012.6491813

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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.
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.
This article develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error-correction performance, high rates, and low decoding complexity. The proposed method produces several infinite families of codes with a wide variety of parameters and entanglement requirements. Our framework encompasses the previously known entanglement-assisted quantum LDPC codes having the best error-correction performance and many other codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.
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 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.