## 3 The RQTM Model  
### Quantum Walk Simulation of Random Exit Selection  
Suppose \(G=(V,E)\) is the road network graph without considering its width, where \(V\) is the set of nodes, which represents the possible choices at each exit, and \(E\) is the set of edges, representing the possible evolution path of random exit selection over time. The number of nodes and edges in G are \(|V|\) and \(|E|\), respectively. \(A\) is the adjacency matrix of network \(G\), that is  
\[A_{uv}=\left\{ \begin{array}{l}1,if(u,v)\in E\\ 0,otherwise\end{array} \right. \tag{5}\]  
Since different exits are connected by the expressway, the moving of drivers in the expressway can be modeled as transitions among different exits. At any time, a binary state indicating whether a driver will or will not exit the expressway (i.e., select or not select a node) can be constructed for each node. Since a driver will pick the travel route from multiple available routes based on considerations of economy, timeliness, and traffic conditions, although a driver has a deterministic destination, the exit that he or she selects can be random due to the multiplicity of available routes. The randomness of the exit selection can be modelled as the probability distribution of the nodes in the decision space. Before we formulated the quantum walk, we first defined a Hilbert space \(H\in\mathbb{C}^{N}\), which is made up of a base state \(|v\rangle\). The characteristic of a Hilbert space \(H\) is that any quantum superposition state is a linear combination of base states, and the superposition state is also in \(H\). Therefore, the state \(|\varphi(k)\rangle\) of any driver with a quantum walk on \(G\) can be defined as a linear superposition state of all base states, which is expressed as follows:  
\[|\varphi(k)\rangle=\sum_{v\in V}a_{v}(k)|v\rangle \tag{6}\]  
where \(a_{v}(k)\) is the probability amplitude of the corresponding base state \(|v\rangle\) at time \(T_{k}\), \(a_{v}(k)\in C\), and \(|a_{v}(k)|\in[0,1]\). The probability of finding the walker at any exit at time \(T_{k}\) is given by the squared modulus of the appropriate element of \(\varphi(k)\)[36]. That is, the product  
Figure 2: The overall framework. of the probability amplitude and its complex conjugate is equivalent to the probability of the walker appearing on a node in the classical space. Therefore, the probability that a driver is in a base state \(|v\rangle\) at time \(T_{k}\) can be expressed as  
\[p(|v\rangle,k)=\alpha_{v}(k)\alpha_{v}^{*}(k) \tag{7}\]  
where \(\alpha_{v}^{*}(k)\) is a complex conjugate of \(\alpha_{v}(k)\). At time \(T_{k}\), it satisfies \(\sum\limits_{v\in V}p(|v\rangle,k)=1\). Unlike the classical random walk, the process of the quantum walk is not a Markov chain [33]. The evolution of state vector \(|\varphi(k)\rangle\) with time \(T_{k}\) is realized by the following unitary transformation:  
\[\frac{d}{dt}|\varphi(k)\rangle=iA|\varphi(k)\rangle \tag{8}\]  
It can be seen from Equation (8) that the state evolution process depends on the adjacency matrix \(A\). However, as a differential equation with complex coefficients, it is difficult to solve it directly. Therefore, it can be transformed into a matrix form from the discrete point of view, and the quantum walk can be efficiently solved by matrix operation. Assuming a driver starts at the node \(|v\rangle\), from Equation (8), we know that it will have a state vector  
\[|\varphi(k)\rangle=e^{-iA}|v\rangle \tag{9}\]  
Then the probability \(p_{vu}(k)\), indicating the walker moves from node \(v\) to node \(u\), is  
\[p_{vu}(k)=||\mu||\varphi(k)\rangle|^{2} \tag{10}\]  
Analogously to Equation (3), we can express the state vector as a matrix  
\[\left( \begin{array}{c}\varphi_{1}{}^{\prime}\\ \vdots\\ \varphi_{n}{}^{\prime}\end{array} \right)=\left( \begin{array}{cccc}\mu_{11} &\ldots&\mu_{1n}\\ \vdots&\ddots&\vdots\\ \mu_{n1}&\ldots&\mu_{nn}\end{array}