Research On The Solvability Of Quantum Multi-unicast Network

With the rapid development of quantum communication, networking and globalization are an inevitable trend of quantum communication. Quantum networks certainly will bring new feasibility to communication systems. How-ever, in the complex network, transmission congestion and channel noises are the main problems influencing the communication efficiency and further the ultimate realization of quantum communication network. In this context, the efficiency of quantum network communication has become a major topic that scholars pay close attention to. Based on this, network coding is introduced to this research area giving birth to a new research direction-Quantum Network Coding (QNC). By combining quantum mechanics with classical network cod-ing theory, the research of quantum network coding potential and limit are the basic theory to promote quantum network communication to practical applica-tion, and seeking the solution with QNC has important theoretical and practical significance to the above problems.The characterization of the solvability of network communication problem is one of the most important problems in the basic theory of network coding.The solvability mainly consider whether a given network can handle a specific communication task. At present, the research on the solvability of quantum networks mainly focuses on two types of communication problems: quantum multicast problem and quantum multiple unicast problem. Since quantum in-formation cannot be cloned (the quantum no-cloning theorem) and quantum network mostly means sending quantum information rather than classical in-formation, multiple unicast networks have been well-studied in several settings and achieve many remarkable results. In this paper, we mainly show the works done in the research of solvability and the prospect of future works focusing on k-pair quantum networks where each source node wishes to communicate with its corresponding sink through a common channel. Firstly, in response to the problem of designing optimization protocols independent of the solvabil-ity of classical network, we give a positive answer and present optimization-al protocol; Secondly, for some network with Lattice structure, we present a specific method to analyze the solvability and introduce coding protocol with lower communication cost; Lastly, for general k-pair network, we present the sufficient conditions of solvability in the measurement-based?In contrast to the gate-based way, measurement-based QNC can resist multiple noise effects more efficiently. Specifically, we yield following positive results:(1)The dependence of existing QNC schemes on classical network solu-tions is analyzed and an optimized QNC scheme independent of classical solv-ability conditions in decoding is proposed. With error-free quantum channels and classical communication, Kobayashi et al proposed a sufficient condition for solvability of a large class of quantum networks, that is, the corresponding classical networks are solvable. In this paper, we have carried out a thorough re-search on whether QNC can eliminate the dependence on classical network and optimize the cost of coding, and give positive conclusions. The study found that certain types of quantum operation are sufficient to support multiple unicast sessions over the extended butterfly network Gk by the idea of graph coloring theory to divide all nodes, so as to optimize the operation resource consump-tions. In addition, this division reduces the number of introduced quantum systems, thus reducing the cost of information storage. n particular, because the partitioning of nodes reduces the encoding space dimension, so that the transmission congestion problem is avoided in the decoding strategy, so the transmission of classical information in QNC is no longer dependent on the classical network solvability. Based on the above design idea, an optimized QNC protocol is proposed. The analysis shows that the protocol has lower communication costs.(2) We mainly concern the coding feasibility of quantum multi-unicast problem over directed acyclic network with mesh structure and a new approach based on stabilizer formalism and the feasibility theorem for simulating quan-tum circuit is presented. By fully considering the network topology as well as its equivalent characterization-2 dimension and 3 dimension cluster states,and reducing the communication problem to some specific quantum operation-swapping operation and identity operation, the problem of determining the solvability of several unicast networks is solved. Firstly, a universal scheme for the generation of resource states among distant communication nodes is provided. The corresponding between cluster and bigraph leads to a constant temporal resource cost. Secondly, for 2-pair networks, we analyze the fea-sibility to simulating the swap operation with stabilizer equations and Pauli measurement model as well as present the specific coding protocol. Further, by 3 dimension cluster states, we come to the conclusion of a 3 dimension butter-fly network. Also, the analysis reveals that the resource consumption involving spatial resources, operational resources and temporal resources mostly reach the lower bounds.(3) We mainly consider the solvability of general quantum k-pair problem using measurement-based elements as well as the advantage of measurement-based QNC compared with gate-based ways. Gate-based QNC by simulating the corresponding classical linear codes are proved to be a measurement-based procedure for which no solvability conditions are presented. In the absence of noise, we present sufficient conditions which build an unambiguous func-tional relationship between the solvability and the structure of network by its equivalent characterization-graph states. This conclusion can also analyze the feasibility of the communication task to sharing k EPR pairs in long-range net-works. Further, in the presence of noise and imperfections, we analyze the advantage of measurement-based QNC in contrast to gate-based way. By an instance network Gk, we show that measurement-based implementation allows higher error thresholds. Specially, for X error, the error threshold is about 30%in MB-QNC is significantly better than in GB-QNC 10%. And the threshold for Z error is slightly better. This conclusion can be extended to any quantum k-pair network whose corresponding classical counterpart is linearly solvable.