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arXiv:2304.04743 · IEEE International Symposium on Information Theory (ISIT)

Improved Logical Error Rate via List Decoding of Quantum Polar Codes

Anqi Gong, Joseph M. Renes

Published Jul 7, 2024• 1 citation
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Abstract

The successive cancellation list decoder (SCL) is an efficient decoder for classical polar codes with low decoding error, approximating the maximum likelihood decoder (MLD) for small list sizes. Here we adapt the SCL to the task of decoding quantum polar codes and show that it inherits the high performance and low complexity of the classical case, and can approximate the quantum MLD for certain channels. We apply SCL decoding to a novel version of quantum polar codes based on the polarization weight (PW) method, which entirely avoids the need for small amounts of entanglement assistance apparent in previous quantum polar code constructions. When used to find the precise error pattern, the quantum SCL decoder (SCL-E) shows competitive performance with surface codes of similar size and low-density parity check codes of similar size and rate. The SCL decoder may instead be used to approximate the probability of each equivalence class of errors, and then choose the most likely class. We benchmark this class-oriented decoder (SCL-C) against the SCL-E decoder and find a noticeable improvement in the logical error rate. This improvement stems from the fact that the contributions from just the low-weight errors give a reasonable approximation to the error class probabilities. Both SCL-E and SCL-C maintain the complexity O(LN log N) of SCL for code size$N$and list size L.

DOI
10.1109/isit57864.2024.10619465

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The successive cancellation list decoder (SCL) is an efficient decoder for classical polar codes with low decoding error, approximating the maximum likelihood decoder (MLD) for small list sizes. Here we adapt the SCL to the task of decoding quantum polar codes and show that it inherits the high performance and low complexity of the classical case, and can approximate the quantum MLD for certain channels. We apply SCL decoding to a novel version of quantum polar codes based on the polarization weight (PW) method, which entirely avoids the need for small amounts of entanglement assistance apparent in previous quantum polar code constructions. When used to find the precise error pattern, the quantum SCL decoder (SCL-E) shows competitive performance with surface codes of similar size and low-density parity check codes of similar size and rate. The SCL decoder may instead be used to approximate the probability of each equivalence class of errors, and then choose the most likely class. We benchmark this class-oriented decoder (SCL-C) against the SCL-E decoder and find a noticeable improvement in the logical error rate. This improvement stems from the fact that the contributions from just the low-weight errors give a reasonable approximation to the error class probabilities. Both SCL-E and SCL-C maintain the complexity O(LN log N) of SCL for code sizeNNNand list size L.
Successive cancellation list (SCL) decoding algorithm for polar codes is one of the best decoding algorithms due to its excellent decoding performance. The decoding complexity of conventional SCL decoding algorithm is O(LN log N). The decoding complexity will increase as the size of the list increases. To solve this problem, we introduce an improved SCL decoding algorithm based on the sequential probability ratio test (SPRT). The proposed algorithm use double thresholds to decode the polar codes instead of single threshold decision used by the conventional SCL algorithm. Numerical results have shown that for (1024, 512) polar code, the proposed SPRT-SCL decoder can achieve a reduction in decoding complexity, and keep the similar decoding performance compared with conventional SCL decoder.
Successive cancellation list (SCL) decoding algorithm is a powerful method that can help polar codes achieve excellent error-correcting performance. However, the current SCL algorithm and decoders are based on likelihood or log-likelihood forms, which render high hardware complexity. In this paper, we propose a log-likelihood-ratio (LLR)-based SCL (LLR-SCL) decoding algorithm, which only needs half the computation and storage complexity than the conventional one. Then, based on the proposed algorithm, we develop low-complexity VLSI architectures for LLR-SCL decoders. Analysis results show that the proposed LLR-SCL decoder achieves 50% reduction in hardware and 98% improvement in hardware efficiency.
Polar codes are capacity achieving error correcting codes that can be decoded through the successive-cancellation algorithm. To improve its error-correction performance, a list-based version called successive-cancellation list (SCL) has been proposed in the past, that however substantially increases the number of time-steps in the decoding process. The simplified SCL (SSCL) decoding algorithm exploits constituent codes within the polar code structure to greatly reduce the required number of time-steps without introducing any error-correction performance loss. In this paper, we propose a faster decoding approach to decode one of these constituent codes, the Rate-1 node. We use this Rate-1 node decoder to develop Fast-SSCL. We demonstrate that only a list-size-bound number of bits needs to be estimated in Rate-1 nodes and Fast-SSCL exactly matches the error-correction performance of SCL and SSCL. This technique can potentially greatly reduce the total number of time-steps needed for polar codes decoding: analysis on a set of case studies show that Fast-SSCL has a number of time-steps requirement that is up to 66.6% lower than SSCL and 88.1% lower than SCL.
Polar codes are capacity achieving error correcting codes that can be decoded through the successive-cancellation algorithm. To improve its error-correction performance, a list-based version called successive-cancellation list (SCL) has been proposed in the past, that however substantially increases the number of time-steps in the decoding process. The simplified SCL (SSCL) decoding algorithm exploits constituent codes within the polar code structure to greatly reduce the required number of time-steps without introducing any error-correction performance loss. In this paper, we propose a faster decoding approach to decode one of these constituent codes, the Rate-1 node. We use this Rate-1 node decoder to develop Fast-SSCL. We demonstrate that only a list-size-bound number of bits needs to be estimated in Rate-1 nodes and Fast-SSCL exactly matches the error-correction performance of SCL and SSCL. This technique can potentially greatly reduce the total number of time-steps needed for polar codes decoding: analysis on a set of case studies show that Fast-SSCL has a number of time- steps requirement that is up to 66.6% lower than SSCL and 88.1% lower than SCL.
For polar codes, successive cancellation list (SCL) decoding algorithm significantly improves finite-length performance compared to SC decoding. SCL-flip decoding can further enhance the performance but the gain diminishes as code length increases, due to the difficulty in locating the first error bit position. In this work, we introduce an SCL-perturbation decoding algorithm to address this issue. A basic version of the algorithm introduces small random perturbations to the received symbols before each SCL decoding attempt, and exhibits non-diminishing gain at large block lengths. Its enhanced version adaptively performs random perturbations or directional perturbation on each received symbol according to previous decoding results, and managed to correct more errors with fewer decoding attempts. Extensive simulation results demonstrate stable gains across various code rates, lengths and list sizes. To the best of our knowledge, this is the first SCL enhancement with non-diminishing gains as code length increases, and achieves unprecedented efficiency. With only one additional SCL-LLL decoding attempt (in total two), the proposed algorithm achieves SCL-2L2L2L-equivalent performance. Since the gain is obtained without increasing list size, the algorithm is best suited for hardware implementation.
Polar codes are the first provably capacity-achieving novel channel codes. Successive cancellation list (SCL) decoding algorithm has a performance close to maximum-likelihood decoding when code length is finite, though the complexity is high. To reduce the complexity of SCL, a partial path expansion with segmented check and pruning SCL (PPE-SCP-SCL) algorithm is proposed in this paper. Based on the successive decoding structure, if the decoding result of a bit is judged to be reliable enough, then a hard decision is made directly and no path expansion is necessary. Moreover, several parity-check points are introduced to perform segmented parity-check during the decoding process, and if the check fails, the current path is directly abandoned. The rationality of the operations is theoretically verified, and numerical simulations also verify that the proposed algorithm can significantly reduce the complexity of decoding with trivial loss of error performance.