# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 1 Introduction  
In recent years, rapid progress has been made in the research and development of standalone quantum computers [1], both in terms of software and hardware, to the point where it is debated whether quantum computational supremacy has achieved [2; 3; 4]. In the near future, standalone quantum computers are expected to show innovative performance in various fields, such as machine learning [5; 6; 7; 8; 9] and computational chemistry [10; 11; 12; 13; 14]. On the other hand, it is known that much quantum information processing, including various different types of quantum cryptography such as quantum public-key cryptography [15], quantum blind computation [16; 17], and quantum money [18; 19] cannot be realized on standalone quantum computers but only on quantum networks [20], where a quantum network of a large size is called quantum internet [21; 22]. Thus, recently, quantum network has been intensively studied both theoretically [23; 24; 25; 26; 27; 28; 29] and experimentally [30; 31]. Theoretical research for improving the throughput of quantum networks started from the study of quantum repeaters aiming to share a maximally entangled state between end vertices of a quantum network represented by a path graph [32; 33]. After an enormous amount of research in this direction (see [20] and references therein), the ultimate limit for sharing a maximally entangled state between a given pair of nodes on a quantum network was finally derived by [PERSON] et al. [24; 26]. On the other hand, since multiple users may simultaneously communicate with each other, it is also important to study multiparty communication protocol on a quantum network. In general, the problem of finding a better multiparty communication protocol on a quantum network is much more complicated than a problem of a two-party communication protocol. First, even on the classical network, there are many different types of multiparty communication. Simple examples ofmultiparty communication may be multiple-unicast communication and multicast communication, where the definitions of these schemes are given later in this introduction. However, in general, an arbitrary type of communication among multiple parties can be considered [34], and we need to optimize the communication protocol depending on the type. Furthermore, there is an additional problem with multiparty communication on quantum networks. As is well known, \"a class of maximally entangled states\" is not unique [35] in a multipartite system, which is represented by the incomparability of GHZ states and W states under stochastic local operation and classical communication (SLOCC) [36]. This fact suggests that depending on our purpose, we need to use a different multiparty protocol to share a different type of states. In other words, we cannot discuss the optimality of the multiparty protocol before we determine what type of states we want to share. In classical information theory, a technique called network coding is known to be useful to improve throughput of various different multiparty communication schemes when there is a bottleneck on a network. Here, network coding is a technique of applying nontrivial operations to the bitstream at intermediate nodes [37; 38; 39]. The method of network coding can be also applied to quantum networks, and quantum information processing on a quantum network that utilizes methods of network coding is called \"a quantum network coding\" [40]. There has been a considerable amount of research on quantum network coding, which tries to improve the throughput of a quantum network in various situations [41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57]. Recently, it has been presented that quantum network coding can improve the security of a quantum network [58; 59; 60; 61] and reduce decoherence effect [62]. Furthermore, it is useful for quantum repeater networks [63; 64] as well as for distributed quantum computation [65]. Moreover, a simple quantum network code has been experimentally demonstrated [66; 67].