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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Improved Ising Model Formulation for Polar Codes

Ryan Seah, Warren J. Gross

Published May 19, 2026• 0 citations
Publisher

Abstract

This paper presents an improved Ising model framework for polar codes, termed POLARIS, which reduces the number of binary variables by incorporating rate-1 node structures and embedding elements of successive-cancellation decoding into the Ising formulation. The decoder scales efficiently to block lengths up to N = 64, doubling prior Ising-based limits. POLARIS achieves near-successive-cancellation list performance within 0.4 dB while reducing QUBO dimensionality from 192 to 126 variables. These advancements bring Ising-based polar decoding closer to practical realization, offering improved efficiency for implementation on both quantum and hybrid CMOS-classical annealing hardware.

DOI
10.1109/ismvl68998.2026.00034

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This paper presents an improved Ising model framework for polar codes, termed POLARIS, which reduces the number of binary variables by incorporating rate-1 node structures and embedding elements of successive-cancellation decoding into the Ising formulation. The decoder scales efficiently to block lengths up to N = 64, doubling prior Ising-based limits. POLARIS achieves near-successive-cancellation list performance within 0.4 dB while reducing QUBO dimensionality from 192 to 126 variables. These advancements bring Ising-based polar decoding closer to practical realization, offering improved efficiency for implementation on both quantum and hybrid CMOS-classical annealing hardware.
The successive-cancellation list (SCL) and successive-cancellation flip (SCF) decoding can be used to improve the performance of polar codes, especially for short to moderate length codes. However, their serial decoding nature results in significant decoding latencies. Implementing some operations in parallel can reduce their decoding latencies. This paper presents fast implementations of the SCL and SCF decoders. In particular, we propose fast parallel list decoders for five newly identified types of nodes in the decoding tree of a polar code, which significantly improves the decoding latency. We also present novel fast SCF decoders that decode some special nodes in the decoding tree of a polar code without serially computing bit log-likelihood ratios. Using our proposed fast parallel SCF decoders, we observed an improvement up to 81% with respect to the original SCF decoder. This significant reduction in the decoding latency is observed without sacrificing the bit-error-rate performance of the code.
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.
This work presents a fast successive-cancellation list flip (Fast-SCLF) decoding algorithm for polar codes that addresses the high latency issue associated with the successive-cancellation list flip (SCLF) decoding algorithm. We first propose a bit-flipping strategy tailored to the state-of-the-art fast successive-cancellation list (FSCL) decoding that avoids tree-traversal in the binary tree representation of SCLF, thus reducing the latency of the decoding process. We then derive a parameterized path selection error model to accurately estimate the bit index at which the correct decoding path is eliminated from the initial FSCL decoding. The trainable parameter is optimized online based on an efficient supervised learning framework. Simulation results show that for a polar code of length 512 with 256 information bits, with similar error-correction performance and memory consumption, the proposed Fast-SCLF decoder reduces up to 73.4% of the average decoding latency of the SCLF decoder with the same list size at the frame error rate of 10−4, while incurring a maximum computational complexity overhead of 27.6%. For the same polar code of length 512 with 256 information bits and at practical signal-to-noise ratios, the proposed decoder with list size 4 reduces 89.3% and 43.7% of the average complexity and decoding latency of the FSCL decoder with list size 32 (FSCL-32), respectively, while also reducing 83.2% of the memory consumption of FSCL-32. The significant improvements of the proposed decoder come at the cost of 0.07 dB error-correction performance degradation compared with FSCL-32.
The soft cancellation decoding of polar codes achieves a better performance than the belief propagation decoding with lower computational time and space complexities. However, because the soft cancellation decoding is based on the successive cancellation decoding, the decoding efficiency and performance with finite-length blocks can be further improved. Exploiting the idea of the successive cancellation list decoding, the soft cancellation decoding can be improved in two aspects: one is by adding branch decoding to the error-prone information bits to increase the accuracy of the soft information, and the other is through using partial iterative decoding to reduce the time and computational complexities. Compared with the original method, the improved soft cancellation decoding makes progress in the error correction performance, increasing the decoding efficiency and reducing the computational complexity, at the cost of a small increase of space complexity.
We show that successive cancellation list decoding can be formulated exclusively using log-likelihood ratios. In addition to numerical stability, the log-likelihood ratio based formulation has useful properties that simplify the sorting step involved in successive cancellation list decoding. We propose a hardware architecture of the successive cancellation list decoder in the log-likelihood ratio domain which, compared with a log-likelihood domain implementation, requires less irregular and smaller memories. This simplification, together with the gains in the metric sorter, lead to$ 56\%$to137%137\%137%higher throughput per unit area than other recently proposed architectures. We then evaluate the empirical performance of the CRC-aided successive cancellation list decoder at different list sizes using different CRCs and conclude that it is important to adapt the CRC length to the list size in order to achieve the best error-rate performance of concatenated polar codes. Finally, we synthesize conventional successive cancellation decoders at large block-lengths with the same block-error probability as our proposed CRC-aided successive cancellation list decoders to demonstrate that, while our decoders have slightly lower throughput and larger area, they have a significantly smaller decoding latency.