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Acta Physica Sinica

Feedback sum-product decoding of sparse quantum codes for X-Z Pauli channels

Jinye Peng, Yun-Jiang Wang, Xinmei Wang, Bai Bao-Ming

Published Jan 1, 2011• 1 citation
Publisher

Abstract

In this paper, a feedback sum-product decoding algorithm of sparse quantum codes for X-Z Pauli channels is developed. Compared with the previous decoding algorithm, our feedback strategy exploits not just the syndrome but also the values of the frustrated checks on individual qubits of the code and the character of the channel model with the portion of each error to adjust the probability distribution of information nodes. Due to the smart adjustment, our decoding algorithm, on one hand, can break the symmetric degeneracy, and on the other hand, can feed back more useful information to the SPA decoder to help the decoder determine a valid output, thereby significantly improving the decoding ability of the decoder. Moreover, our algorithm, which is based on GF(4), overcomes the limitation caused by decoding in GF(2). Finally, we want to point out that, our method does not increase the measurement overhead in comparison wioth the previous methods, as the extra information comes for free from the requisite stabilizer measurement.

DOI
10.7498/aps.60.030306

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In this paper, a feedback sum-product decoding algorithm of sparse quantum codes for X-Z Pauli channels is developed. Compared with the previous decoding algorithm, our feedback strategy exploits not just the syndrome but also the values of the frustrated checks on individual qubits of the code and the character of the channel model with the portion of each error to adjust the probability distribution of information nodes. Due to the smart adjustment, our decoding algorithm, on one hand, can break the symmetric degeneracy, and on the other hand, can feed back more useful information to the SPA decoder to help the decoder determine a valid output, thereby significantly improving the decoding ability of the decoder. Moreover, our algorithm, which is based on GF(4), overcomes the limitation caused by decoding in GF(2). Finally, we want to point out that, our method does not increase the measurement overhead in comparison wioth the previous methods, as the extra information comes for free from the requisite stabilizer measurement.
Decoding sparse quantum codes can be accomplished by syndrome-based decoding through using the sum-product algorithm (SPA). We significantly improve this decoding scheme by developing a new feedback adjustment strategy for the standard SPA. In our feedback strategy, we exploit not only the syndrome but also the values of the frustrated checks on individual qubits of the code and the channel model. Consequently, our decoding algorithm, on the one hand, can break the symmetric degeneracy, and on the other hand, can feed back more useful information to the SPA decoder to help the decoder determine a valid output, thereby significantly improving the decoding ability of the decoder. Moreover, our algorithm does not increase the measurement complexity compared with the previous method, but takes full advantage of the measured information.
Decoding sparse quantum codes can be accomplished by syndrome-based decoding using a belief propagation (BP) algorithm. We significantly improve this decoding scheme by developing a new feedback adjustment strategy for the standard BP algorithm. In our feedback procedure, we exploit much of the information from stabilizers, not just the syndrome but also the values of the frustrated checks on individual qubits of the code and the channel model. Furthermore we show that our decoding algorithm is superior to belief propagation algorithms using only the syndrome in the feedback procedure for all cases of the depolarizing channel. Our algorithm does not increase the measurement overhead compared to the previous method, as the extra information comes for free from the requisite stabilizer measurements.
Decoding sparse quantum codes can be accomplished by syndrome-based decoding using a belief propagation (BP) algorithm. We significantly improve this decoding scheme by developing a new feedback adjustment strategy for the standard BP algorithm. In our feedback procedure, we exploit much of the information from stabilizers, not just the syndrome but also the values of the frustrated checks on individual qubits of the code and the channel model. Furthermore we show that our decoding algorithm is superior to belief propagation algorithms using only the syndrome in the feedback procedure for all cases of the depolarizing channel. Our algorithm does not increase the measurement overhead compared to the previous method, as the extra information comes for free from the requisite stabilizer measurements.
In this paper, we generalize the feedback sum-product algorithm to the decoding of sparse quantum codes over random balanced X-Z Channels, thereby significantly improving the performance of the concerned sparse quantum codes. We point out that, due to the randomness of the channel, degeneracy errors would be degreased greatly and the improvement of the performance of the sparse quantum codes come from the new decoding algorithm, where more reliable and useful information is feedback to the decoder by using the feedback strategy. As the end of this paper, simulations are given to verify our statements.
For quantum LDPC codes widely studied recent years,the decoding strategy based on the feedback sum-product( FSPA) we proposed before is investigated for the case of unbalanced X-Z channel. And the analysis of the corresponding performance of the FSPA is given. The study shows that under the unbalanced X-Z channel model, the feedback strategy could still exploit the syndrome comparison,the error style of the frustrated checks,the channel character analysis to adjust the probabilities distributions of the concerned information bits,thereby significantly improving the decoding ability of the sum-product decoder. Finally,simulation results verified our statements.
Quantum stabilizer codes constructed from sparse matrices have good performance and can be efficiently decoded by belief propagation (BP). A conventional BP decoding algorithm treats binary stabilizer codes as additive codes over GF(4). This algorithm has a relatively complex process of handling check-node messages, which incurs higher decoding complexity. Moreover, BP decoding of a stabilizer code usually suffers a performance loss due to the many short cycles in the underlying Tanner graph. In this paper, we propose a refined BP decoding algorithm for quantum codes with complexity roughly the same as binary BP. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but the passed node-to-node messages are single-valued, unlike the quaternary BP, where multivalued node-to-node messages are required. Furthermore, the techniques of message strength normalization can naturally be applied to these single-valued messages to improve the performance. Another observation is that the message-update schedule affects the performance of BP decoding against short cycles. We show that running BP with message strength normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error floor in computer simulation.