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1. Belief-Propagation with Quantum Messages for Polar Codes on Classical-Quantum Channels

   Mandal, Avijit; Brandsen; S. Pfister, Henry D. (January 2023, 2023 IEEE International Symposium on Information Theory (ISIT))

   This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results. 

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2. Belief-Propagation with Quantum Messages for Polar Codes on Classical-Quantum Channels, IEEE International Symposium on Information Theory, Taipei, Taiwan, June 25-30, 2023

   https://doi.org/10.1109/ISIT54713.2023.10206723  

   Mandal, Avijit; Brandsen, S; Pfister, Henry D. (June 2023, Abstracts of papers IEEE International Symposium on Information Theory)

   This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results. 

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3. Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided Decimation

   https://doi.org/10.1109/Allerton63246.2024.10735275  

   Gökduman, Mert; Yao, Hanwen; Pfister, Henry D (September 2024, IEEE)

   Quantum low-density parity-check (LDPC) codes are a promising family of quantum error-correcting codes for fault tolerant quantum computing with low overhead. Decoding quantum LDPC codes on quantum erasure channels has received more attention recently due to advances in erasure conversion for various types of qubits including neutral atoms, trapped ions, and superconducting qubits. Belief propagation with guided decimation (BPGD) decoding of quantum LDPC codes has demonstrated good performance in bit-flip and depolarizing noise. In this work, we apply BPGD decoding to quantum erasure channels. Using a natural modification, we show that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes. Furthermore, we show that the performance of BPGD decoding on erasure channels can sometimes be improved significantly by either adding damping or adjusting the initial channel log-likelihood ratio for bits that are not erased. More generally, our results demonstrate BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape. 

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4. On Sparse Regression LDPC Codes

   https://doi.org/10.1109/ISIT54713.2023.10206818  

   Ebert, Jamison R; Chamberland, Jean-Francois; Narayanan, Krishna R (August 2023, IEEE)

   Iterative decoding of graph-based codes and sparse recovery through approximate message passing (AMP) are two research areas that have seen monumental progress in recent decades. Inspired by these advances, this article introduces sparse regression LDPC codes (SR-LDPC codes) and their decoding. Sparse regression codes (SPARCs) are a class of error correcting codes that build on ideas from compressed sensing and can be decoded using AMP. In certain settings, SPARCs are known to achieve capacity; yet, their performance suffers at finite block lengths. Likewise, low-density parity-check (LDPC) codes can be decoded efficiently using belief propagation and can also be capacity achieving. This article introduces a novel concatenated coding structure that combines an LDPC outer code with a SPARC-inspired inner code. Efficient decoding for such a code can be achieved using AMP with a denoiser that performs belief propagation on the factor graph of the outer LDPC code. The proposed framework exhibits performance improvements over SPARCs and standard LDPC codes for finite block lengths and results in a steep waterfall in error performance, a phenomenon not observed in uncoded SPARCs. 

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5. Belief propagation with quantum messages for quantum-enhanced classical communications

   https://doi.org/10.1038/s41534-021-00422-1  

   Rengaswamy, Narayanan; Seshadreesan, Kaushik P.; Guha, Saikat; Pfister, Henry D. (December 2021, npj Quantum Information)

   null (Ed.)

   Abstract For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity achievable with a receiver that performs individual pulse-by-pulse detection, and the quantum-optimal “joint-detection receiver” that acts collectively on long codeword-blocks of modulated pulses; an effect often termed “superadditive capacity”. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a qubit, described by an inner product that is a function of the mean photon number per pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of “belief-propagation with quantum messages” (BPQM). We quantify its performance improvement over the Dolinar receiver that performs optimal pulse-by-pulse detection, which represents the best “classical” approach. We analyze the scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 8 (BPSK) codewords of a length-5 binary linear code with a tree factor graph. Our result suggests that a BPQM receiver might attain the Holevo capacity of this BPSK-modulated pure-loss channel. Moreover, our receiver circuit provides an alternative proposal for a quantum supremacy experiment, targeted at a specific application that can potentially be implemented on a small, special-purpose, photonic quantum computer capable of performing cat-basis universal qubit logic. 

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