**1. Introduction**

#### **1.1 The origin of quantum information**

Quantum mechanics' establishment and development triggered the first wave of quantum technology in the twentieth century. With the regulation and observation of microphysical quantity as the main feature of understanding and grasping the microphysical phenomena and laws, quantum information based on the principles of quantum mechanics was born. Quantum information, a new information method, that calculates, encodes, and transmits the physical information contained in the "state" of a quantum system. The most common unit of quantum information is the qubit, that is, intrinsically linked to each other and can be any combination of 0 and 1 simultaneously.

### **2. The development history of quantum information**

Quantum information technologies aim to use the natural characteristic of the atomic scale to accomplish tasks that cannot be achieved with existing technologies and use the characteristic of measuring or observing a quantum system to change the quantum information fundamentally. These technologies rely on qubits. Meanwhile, scientists are creating physical qubits from a variety of particles, such as atoms or light particles, or objects that mimic them, such as superconducting circuits. Scientists manipulate the quantum properties of each qubit and entangle multiple qubits with each other to create quantum technology from these qubits. These functions support two potential transformative applications, that is, quantum computing and quantum communications. However, quantum information is fragile and can be irreversibly lost through interactions with the environment, a process known as decoherence. Quantum error correction techniques have been proposed and proven, but are challenging to implement. Based on these, researchers began to explore the application of quantum information to quantum technologies in the twentieth century.


In the twenty-first century, the theory and development of quantum computing and communications puts this significance on a firm footing and leads to some new profound and exciting insights into the natural world. From 2000 to 2005, a variety of time-efficient quantum algorithms were proposed, such as the semiproduct groups [8–10], the near-Hamiltonian groups [11], the normal subgroups [12, 13], the almost Abelian groups [14]. In 2006, Hayashi et al. [15] proposed the first quantum network coding scheme, which realized the cross transmission of two qubits in a full quantum channel butterfly network. Due to the constraints of quantum properties, such as the quantum unclonable theorem, this scheme cannot achieve lossless quantum transmission, that is, the fidelity is less than 1. In 2012, Satoh et al. [16] designed a novel quantum network coding scheme using quantum repeaters. In 2014, Nishimura [17] summarized the current state of quantum network coding, discussing the nature of quantum network coding schemes using entangled resources to communicate with classical. In 2020, Wu et al. [18] proposed a continuous-variable quantum network coding scheme based on a butterflyshaped network model.
