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arXiv:1204.4223

Improved Quantum LDPC Decoding Strategies For The Misidentified Quantum Depolarizing Channel

Yixuan Xie, Jun Li, Robert Malaney, Jinhong Yuan

Published Apr 18, 2012• 0 citations
arXivHTMLPDF

Abstract

Quantum cryptography via key distribution mechanisms that utilize quantum entanglement between sender-receiver pairs will form the basis of future large-scale quantum networks. A key engineering challenge in such networks will be the ability to correct for decoherence effects in the distributed entanglement resources. It is widely believed that sophisticated quantum error correction codes, such as quantum low-density parity-check (LDPC) codes, will be pivotal in such a role. However, recently the importance of the channel mismatch effect in degrading the performance of deployed quantum LDPC codes has been pointed out. In this work we help remedy this situation by proposing new quantum LDPC decoding strategies that can significantly reduce performance degradation by as much as $50\%$. Our new strategies for the quantum LDPC decoder are based on previous insights from classical LDPC decoders in mismatched channels, where an asymmetry in performance is known as a function of the estimated channel noise. We show how similar asymmetries carry over to the quantum depolarizing channel, and how an estimate of the depolarization flip parameter weighted to larger values leads to significant performance improvement.

DOI
10.48550/arxiv.1204.4223

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Quantum cryptography via key distribution mechanisms that utilize quantum entanglement between sender-receiver pairs will form the basis of future large-scale quantum networks. A key engineering challenge in such networks will be the ability to correct for decoherence effects in the distributed entanglement resources. It is widely believed that sophisticated quantum error correction codes, such as quantum low-density parity-check (LDPC) codes, will be pivotal in such a role. However, recently the importance of the channel mismatch effect in degrading the performance of deployed quantum LDPC codes has been pointed out. In this work we help remedy this situation by proposing new quantum LDPC decoding strategies that can significantly reduce performance degradation by as much as 50%50\%50%. Our new strategies for the quantum LDPC decoder are based on previous insights from classical LDPC decoders in mismatched channels, where an asymmetry in performance is known as a function of the estimated channel noise. We show how similar asymmetries carry over to the quantum depolarizing channel, and how an estimate of the depolarization flip parameter weighted to larger values leads to significant performance improvement.
In this work, the importance of the channel mismatch effect in degrading the performance of deployed quantum LDPC codes has been pointed out. We help remedy this situation by proposing new quantum LDPC decoding strategies that can significantly reduce performance degradation by as much as 50%. Our new strategies for the quantum LDPC decoder are based on previous insights from classical LDPC decoders in mismatched channels, where an asymmetry in performance is known as a function of the estimated channel noise. We show how similar asymmetries carry over to the quantum depolarizing channel, and how an estimate of the depolarization flip parameter weighted to larger values leads to significant performance improvement.
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.
Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular [[9216,4612,≤48]][[9216,4612, \leq 48]][[9216,4612,≤48]] quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as 10−810^{-8}10−8 on the depolarizing channel with error probability 4%4 \%4%.
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.
Quantum Low Density Generator Matrix (QLDGM) codes based on Calderbank-Steane-Shor (CSS) constructions have shown unprecedented error correction capabilities in the paradigm of quantum communication. Recently, a strategy based on non-CSS quantum codes derived from QLDGM CSS codes has been shown to surpass other Quantum Low Density Parity Check (QLDPC) schemes proposed in the literature over the depolarizing channel. Given the importance of quantum channel estimation and the impact it has on the performance of QLDPC codes, in this article, we study the behaviour of non-CSS QLDGM codes under the umbrella of channel mismatch. We begin by showing how a relatively accurate estimate of the quantum channel is pivotal for these codes to perform appropriately. We follow this by analyzing an off-line and an on-line quantum channel parameter estimation technique, as well as discussing how these methods affect the Quantum Error Correction (QEC) codes under consideration. Finally, we show how the on-line methodology yields similar performance to the perfect channel knowledge scenario despite its relative simplicity.
Quantum low-density parity-check (QLDPC) codes have been considered as a promising solution for the fault-tolerant quantum computing. However, the belief propagation (BP) decoding for QLDPC codes does not take into account the degeneracy, resulting in certain performance degradation. Recently, various post-processing algorithms have been proposed to address this issue but they in return require excessive additional complexity and/or long decoding latency. Motivated by this, in this paper, we propose an efficient decoding algorithm that takes reduced decoding complexity for the post-processing. In particular, the proposed algorithm performs the post-processing in such a way that it first selects a variable node based on a proposed metric, and the depolarizing channel model for the selected variable node is reinitialized according to the BP decoding results. Then, the BP decoding is performed for the depolarizing channel model in which the selected node follows the reinitialized channel characteristics. The process of selection, reinitialization and BP decoding, say a trial, is iterated until it reaches a predetermined value or all syndromes are met. The main ideas of this work lie in formulating the metric for selection and reinitialization to minimize the number of trials until the degeneracy is resolved. Finally, simulation results show that the proposed decoding algorithms can significantly reduce the decoding complexity with similar decoding performance to that of an existing algorithm.