In recent years, with the rapid development of e-commerce and computer, online shopping is more and more popular. In the meantime, the development of quantum algorithms makes the traditional e-payment protocols based on difficult mathematical problems more and more insecure, so the e-payment protocols based on quantum algorithms come into being. At present, most of the proposed quantum e-payment protocols use entangled states for quantum electronic signature protocols. However, the preparation and measurement of entangled states are very difficult, so in the case of ensuring the security of the protocols, using quantum states featuring more convenient preparation and measurement, instead of entangled states, has become a research direction of e-payment. Quantum walk is a technology that can produce the necessary entanglement resources spontaneously only by using the single-particle states without preparing entanglement resources in advance. This technology has been widely used in quantum computing and quantum simulation and is of certain practical value. At the same time, as people pay more attention to personal privacy, only users' shopping lists being confidential to banks have been unable to meet people's privacy needs. Therefore, in order to solve the above problems, we modify a classic e-payment agreement model and make the hidden users' identity information not affect the normal delivery and merchants. Furthermore, we combine the quantum walk with quantum e-payment and propose a quantum e-payment protocol to ensure that entanglement resources can be obtained without preparing entangled states in advance and guarantee that the buyers' bank accounts and real identity information can be kept confidential to merchants.

Quantum walk is an extension of random walk in the quantum field. It takes the quantum as the carrier to simulate the chaotic nonlinear dynamic walk behavior. According to the characteristics of the quantum walk, the encrypted quantum communication channels are established accordingly. The quantum walk mainly contains the complex Hilbert space of two main quantum spaces, namely, coin space and position space. The protocol is based on the one-dimensional quantum walk teleportation, and through the two-step quantum walk, the shift operator can make the position space and coin space entangle with each other, so as to construct quantum channels for information transmission. The biggest advantage of quantum walk technology is that it can obtain the entanglement resources through the single photon operation. The measurement and preparation of the protocol are simpler compared with directly operating entangled states, and the randomness of the quantum walk makes the transmission more secure. In addition, by dividing the shopping information of buyers into the identity information accessible to the banks and making the shopping list open to the merchants, the banks and the merchants in the protocol will not know the information obtained by the other party so that the privacy of the buyers is greatly protected.

Firstly, in order to further protect the users' privacy, we modify a classic electronic payment agreement model (Fig. 1). In this model, we reduce the workload of buyers and give the processing and distribution of information to third-party platforms. While keeping the buyers' shopping lists confidential to the banks, the buyers' identity information is also unavailable to the merchants. So this protocol makes the merchants and the banks only have the information that they need and know nothing of the information obtained by the other party. Secondly, quantum walk technology is applied to various stages of the protocol including the trading purchase phase, trading payment phase, and verifying phase (Fig. 2). By applying quantum walk technology, the complexity of quantum resource preparation and measurement in the protocol is reduced. Finally, the security analysis of this protocol is conducted (Fig. 3), and the result shows that neither the internal nor external attackers of this protocol can obtain the secret information in the protocol, and this protocol can resist both internal and external attacks.

This protocol, compared with the existing quantum e-payment protocols, not only retains the third-party platforms and the inter-bank payment function but also combines the quantum walk and electronic payment. It makes the participants of the protocol in the initial stage free from preparing particles in entangled states and makes them only prepare the single-particle states which can get the required entanglement resources. This move reduces the complexity of quantum state preparation and measurement. At the same time, the shopping information of the buyers is divided, with the information of the purchased goods confidential to the banks and the buyers' private information unavailable to the merchants. In addition, when the merchants really need this part of the information of the buyers, they can apply to third-party platforms for the information. Being reviewed by the platforms and approved by the buyers, the merchants can know the information they want, so as to complete the corresponding operation. Finally, security analysis shows that this protocol can resist internal and external attacks and is safe and feasible under current technology.