Classical locally recoverable codes (LRCs) have become indispensable in distributed storage systems. They provide efficient recovery in terms of localized errors. Quantum LRCs have very recently been introduced for th...
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Classical locally recoverable codes (LRCs) have become indispensable in distributed storage systems. They provide efficient recovery in terms of localized errors. Quantum LRCs have very recently been introduced for their potential application in quantum data storage. In this paper, we use classical LRCs to investigate quantum LRCs. We prove that the parameters of quantum LRCs are bounded by their classical counterparts. We deduce bounds on the parameters of quantum LRCs from bounds on the parameters of the classical ones. We establish a characterization of optimal pure quantum LRCs based on classical codes with specific properties. Using well-crafted classical LRCs as ingredients in the construction of quantum css codes, we offer the first construction of several families of optimal pure quantum LRCs.
We investigate css and css-T quantum error-correcting codes from the point of view of their existence, rarity, and performance. We give a lower bound on the number of pairs of linear codes that give rise to a css code...
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We investigate css and css-T quantum error-correcting codes from the point of view of their existence, rarity, and performance. We give a lower bound on the number of pairs of linear codes that give rise to a css code with good correction capability, showing that such pairs are easy to produce with a randomized construction. We then prove that css-T codes exhibit the opposite behaviour, showing also that, under very natural assumptions, their rate and relative distance cannot be simultaneously large. This partially answers an open question on the feasible parameters of css-T codes. We conclude with a simple construction of css-T codes from Hermitian curves. The paper also offers a concise introduction to css and css-T codes from the point of view of classical coding theory.
Quantum key distribution (QKD) allows authenticated parties to share secure keys. Its security comes from quantum physics rather than computational complexity. The previous work has been able to demonstrate the securi...
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Quantum key distribution (QKD) allows authenticated parties to share secure keys. Its security comes from quantum physics rather than computational complexity. The previous work has been able to demonstrate the security of the BB84 protocol based on the uncertainty principle, entanglement purification and information theory. In the security proof method based on entanglement purification, it is assumed that the information of Calderbank-Shor-Steane (css) error correction code cannot be leaked, otherwise, it is insecure. However, there is no quantitative analysis of the relationship between the parameter of css code and the amount of information leaked. In the attack and defense strategy of the actual quantum key distribution system, especially in the application of the device that is easy to lose or out of control, it is necessary to assess the impact of the parameter leakage. In this paper, we derive the relationship between the leaked parameter of css code and the amount of the final key leakage based on the BB84 protocol. Based on this formula, we simulated the impact of different css code parameter leaks on the final key amount. Through the analysis of simulation results, the security of the BB84 protocol is inversely proportional to the value of n - k(1) and k(1) - k(2) in the case of the css code leak.
We give a construction of quantum LDPC codes of dimension Theta(logN) and distance Theta(N/logN) as the code length N -> infinity. Using a product of chain complexes this construction also provides a family of quan...
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We give a construction of quantum LDPC codes of dimension Theta(logN) and distance Theta(N/logN) as the code length N -> infinity. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance Omega(N1-alpha/2/logN) and dimension Omega(N-alpha logN), where 0 <= alpha < 1. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixedR < 1 there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least R with, in some sense, optimal circulant size Omega(N/logN) as the code length N -> infinity.
The stowage plan and the securing arrangement of non-standardized cargo are some of the most important aspects in terms of cargo safety and economic costs. For this reason, the optimization of this operation is crucia...
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The stowage plan and the securing arrangement of non-standardized cargo are some of the most important aspects in terms of cargo safety and economic costs. For this reason, the optimization of this operation is crucial and a daily challenge for securing planners trying to fulfill both requirements. In the present paper, a new methodological optimization is proposed presenting novel mathematical models and new 3D maps useful for the people in charge of stowage and securing arrangement. For this goal, the materials followed were the code of Safe Practice for Cargo Stowage and Securing (css) of the International Maritime Organization (IMO) because many ships' Cargo Securing Manuals refer to this international standard. Using an initial case study and making use of response surface techniques of design of experiments (DOE), multiple numerical simulations were performed to obtain novel mathematical models, revealing a high precision. Moreover, new 3D maps were presented and are very interesting tools due to their ease of understanding. The obtained results were compared with other simulations carried out where different variables were employed. The presented models of this methodology can predict the best securing arrangement to fulfill the balance forces in any stowage position (vertical and longitudinal) using the minimum securing devices and keeping the standards of safety. These methodological tools offer valuable advice to the shipping industry with responsibilities involved in the securing design of non-standardized cargo items.
In this article, we define homological quantum codes in arbitrary qudit dimensions D = 2 by directly defining css operators on a 2-Complex S. If the 2-Complex is constructed from a surface, we obtain a qudit surface c...
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In this article, we define homological quantum codes in arbitrary qudit dimensions D = 2 by directly defining css operators on a 2-Complex S. If the 2-Complex is constructed from a surface, we obtain a qudit surface code. We then prove that the dimension of the code we define always equals the size of the first homology group of S. We also define the distance of the codes in this setting, finding that they share similar properties with their qubit counterpart. Additionally, we generalize the hypermap-homology quantum code proposed by Martin Leslie to the qudit case. For every such hypermap code, we construct an abstract 2-Complex whose homological quantum code is equivalent to the hypermap code.
We consider the secure quantum communication over a network with the presence of a malicious adversary who can eavesdrop and contaminate the states. The network consists of noiseless quantum channels with the unit cap...
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We consider the secure quantum communication over a network with the presence of a malicious adversary who can eavesdrop and contaminate the states. The network consists of noiseless quantum channels with the unit capacity and the nodes which applies noiseless quantum operations. As the main result, when the maximum number $m_code$ of the attacked channels over the entire network uses is less than a half of the network transmission rate $m_code$ (i.e., $m_code < m_code/2$ ), our code implements secret and correctable quantum communication of the rate $m_code-2m_code$ by using the network asymptotic number of times. Our code is universal in the sense that the code is constructed without the knowledge of the specific node operations and the network topology, but instead, every node operation is constrained to the application of an invertible matrix to the basis states. Moreover, our code requires no classical communication. Our code can be thought of as a generalization of the quantum secret sharing.
We propose the construction and error correction procedures of an entanglement assisted binary quantum tensor product code. We devise an efficient procedure to construct the code with complexity O(max{rho(2)(1), rho(2...
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ISBN:
(纸本)9781509030972
We propose the construction and error correction procedures of an entanglement assisted binary quantum tensor product code. We devise an efficient procedure to construct the code with complexity O(max{rho(2)(1), rho(2)(2)}) compared to O(rho(2)(1), rho(2)(2)) using symplectic Gram-Schmidt orthogonalization, where rho(1) and rho(2) are the number of parity bits of the component codes. Our error correction procedures can correct quantum burst errors without destroying the quantum state.
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