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Zero-error Coding Method Based on Quantum Five-symbol Confusion Channel Model

LI Jingya,YU Wenbin,LIU Wenjie,WANG Jinwei

Published Dec 1, 2018• 0 citations
Publisher

Abstract

In view of the lack of effective coding schemes for quantum zero-error channel,based on the characteristics of quantum five-symbol confusion channel model and the theory of matrix theory,a coding method of five-symbol confusion channel combining quantum superposition state zero-error coding is proposed.In order to improve the channel capacity and reduce the algorithm complexity,zero-error coding is performed by using the isomorphism relationship between quantum superposition states and vectors and between channels and matrices.Analysis results show that this method has higher channel capacity and coding efficiency than the classical confusion channel coding method.

DOI
10.19678/j.issn.1000-3428.0051106

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In view of the lack of effective coding schemes for quantum zero-error channel,based on the characteristics of quantum five-symbol confusion channel model and the theory of matrix theory,a coding method of five-symbol confusion channel combining quantum superposition state zero-error coding is proposed.In order to improve the channel capacity and reduce the algorithm complexity,zero-error coding is performed by using the isomorphism relationship between quantum superposition states and vectors and between channels and matrices.Analysis results show that this method has higher channel capacity and coding efficiency than the classical confusion channel coding method.
The existing noise channel coding methods need to be optimised in terms of channel capacity and algorithm complexity. Based on the characteristics of the quantum 5-symbol confusion-channel model and the theory of matrices, a coding method is proposed which combines the 5-symbol confusion channel with the quantum-superposition-state coding. This coding method uses two isomorphism steps to obtain the zero-error code words, which are the isomorphism between quantum superposition states and vectors and the isomorphism between channels and matrices. The theoretical derivation proves that the channel capacity is able to increase by employing quantum zero-error coding other than its classical counterpart.
The existing noise channel coding methods need to be optimised in terms of channel capacity and algorithm complexity. Based on the characteristics of the quantum 5-symbol confusion-channel model and the theory of matrices, a coding method is proposed which combines the 5-symbol confusion channel with the quantum-superposition-state coding. This coding method uses two isomorphism steps to obtain the zero-error code words, which are the isomorphism between quantum superposition states and vectors and the isomorphism between channels and matrices. The theoretical derivation proves that the channel capacity is able to increase by employing quantum zero-error coding other than its classical counterpart.
Today's sensor networks need robustness, security and efficiency with a high level of assurance. Error correction is an effective communicational technique that plays a critical role in maintaining robustness in informational transmission. The general way to tackle this problem is by using forward error correction (FEC) between two communication parties. However, by applying zero-error coding one can assure information fidelity while signals are transmitted in sensor networks. In this study, we investigate zero-error coding via both classical and quantum channels, which consist of n obfuscated symbols such as Shannon's zero-error communication. As a contrast to the standard classical zero-error coding, which has a computational complexity of , a general approach is proposed herein to find zero-error codewords in the case of quantum channel. This method is based on a n-symbol obfuscation model and the matrix's linear transformation, whose complexity dramatically decreases to . According to a comparison with classical zero-error coding, the quantum zero-error capacity of the proposed method has obvious advantages over its classical counterpart, as the zero-error capacity equals the rank of the quantum coefficient matrix. In particular, the channel capacity can reach n when the rank of coefficient matrix is full in the n-symbol multilateral obfuscation quantum channel, which cannot be reached in the classical case. Considering previous methods such as low density parity check code (LDPC), our work can provide a means of error-free communication through some typical channels. Especially in the quantum case, zero-error coding can reach both a high coding efficiency and large channel capacity, which can improve the robustness of communication in sensor networks.
This paper introduces a fundamentally new method of finding the conditions for the superactivation of the zero-error capacity of quantum channels.The zero-error capacity of the quantum channel describes the amount of information which can be transmitted perfectly through a noisy quantum channel.The superactivation of the zero-error capacity of quantum channels makes it possible to use two quantum channels, each with zero zero-error capacity, with a positive joint zero-error capacity.Currently we have no theoretical background for describing all possible combinations of superactive quantum channels, hence there should be many other possible combinations.We give an algorithmic solution to the problem.To analyze the superactivation of the zero-error capacity, we introduce a new geometrical representation, called the quantum superball.Our method can be the first efficient algorithmic solution to discover the still unknown combinations to determine the superactivation of the zero-error capacity of quantum channels, without the extremely high computational costs.
This paper presents a novel approach to quantum error correction code design based on information theory principles. The core claim is that by leveraging the foundations of channel capacity theory and coding theory, we can create codes specifically tailored to the constraints and opportunities presented by quantum hardware. The proposed mechanism utilizes these theories to maximize information transmission efficiency and error correction capabilities, resulting in a code that is both robust and amenable to implementation within current and future quantum devices. The research addresses a critical limitation of traditional error correction codes, which often lack the nuanced understanding necessary to effectively mitigate the unique challenges posed by quantum systems. The resulting code offers a significant improvement in the reliability of quantum information transmission, a vital step towards realizing practical quantum communication and computation. This work focuses on the theoretical design and analysis of the code, without the inclusion of experimental results. ---
We study the use of quantum entanglement in the zero-error source-channel coding problem. Here, Alice and Bob are connected by a noisy classical one-way channel, and are given correlated inputs from a random source. Their goal is for Bob to learn Alice's input while using the channel as little as possible. In the zero-error regime, the optimal rates of source codes and channel codes are given by graph parameters known as the Witsenhausen rate and Shannon capacity, respectively. The Lovász theta number, a graph parameter defined by a semidefinite program, gives the best efficiently computable upper bound on the Shannon capacity and it also upper bounds its entanglement-assisted counterpart. At the same time, it was recently shown that the Shannon capacity can be increased if Alice and Bob may use entanglement. Here, we partially extend these results to the source-coding problem and to the more general source-channel coding problem. We prove a lower bound on the rate of entanglement-assisted source-codes in terms of Szegedy's number (a strengthening of the theta number). This result implies that the theta number lower bounds the entangled variant of the Witsenhausen rate. We also show that entanglement can allow for an unbounded improvement of the asymptotic rate of both classical source codes and classical source-channel codes. Our separation results use low-degree polynomials due to Barrington, Beigel and Rudich, Hadamard matrices due to Xia and Liu, and a new application of remote state preparation.