# Four-Qubit Cluster States Generation through Multi-Coin Quantum Walk
## 1 Introduction  
With the development of quantum information technology, quantum computing is becoming more demanding and realistic than ever. Traditional quantum computing is based on quantum circuits composed of single-qubit and multi-qubit quantum gates, which is difficult to implement in real world. As an improvement of traditional quantum computing, [PERSON] and [PERSON] et al. proposed the concept of measurement-based quantum computing (MBQC) in 2001 [1, 2]. Although MBQC simplifies the process of quantum computing, it requires an efficient way to generate cluster states to carry out the operations.  
Cluster states, introduced by [PERSON] in 2004 [3], are multiple qubits that are highly entangled. Compared with other types of entangled states, cluster states are easier to maintain their entangled properties, which is very suitable for quantum computing. Cluster states are traditionally generated based on optical systems and requires a large number of repetitive operations. Improved methods have been proposed: [PERSON] et al. proposed the Browne-Rudolph protocol in 2005 [4], which can generate new cluster states with a probability of 0.5 by using two types of fusion gates while consuming one photon; [PERSON] et al. proposed an improved Browne-Rudolph protocol in 2006 [5]; [PERSON] et al. proposed a new method for generating cluster states using weak non-linearity in 2007 [6]. However, these methods are still too complicated to be implemented in real applications. Therefore, easier and more efficient methods of cluster state generation are required.  
Multi-coin quantum walk can be used to address this problem. Coined quantum walk is one kind of discrete-time quantum walk that combines the freedom of qubits with the classical random walk and defines a new type of quantum coin [7, 8]. Additionally, multi-coin quantum walk involves several coin operators that manipulate the qubits, which can be implemented to carry out specific operations [9, 10, 11, 12]. Recent researches have proved that quantum walk-like protocols can be used to carry out entanglement swapping [13] and multi-coin quantum walk can generate 2-qubit entangled state and 3-qubit GHZ state using only Bell states and local measurement [14]. It has also been proved that coinedquantum walk can be used to construct quantum gates and implement universal computing in 2-qubit ot 3-qubit systems [15]. In this paper, we extend the method proposed in [14] and generate 4-qubit cluster states between targeted qubits with 3 pairs of Bell states and local measurement, which significantly simplifies the process of traditional cluster state generation. We also propose a practical implementation of 4-qubit system of quantum gates and construct a quantum circuit using multi-coin quantum walk to generate 4-qubit cluster state. This method provides a more physically feasible option for real applications.  
This paper is organized as follows. First, we briefly introduce the process of multi-coin quantum walk and describe the generation of 3-qubit GHZ state and 4-qubit cluster state in detail. Next, we introduce the formation of Controlled-Z gate and Hadamard gate in a 4-qubit system using multi-coin quantum walk and then generate 4-qubit cluster state with the composed quantum circuit. Finally, we make a summary of our work.