## 2 Problem Definition and Basic Idea
### Formal Definition of the Problem
Procedure 2: Modeling the probability evolution of exit selection. The dynamics of exit selection can be formulated as a dynamic path generation according to the dynamic change of the probability vector. Typically, the dynamic change of the probability vector can be expressed by the state transition matrix, namely,  
\[P(i,j,k)_{s1}=T_{s0\to s1}P(i,j,k)_{s0} \tag{2}\]  
The full matrix expression form of Equation (2) is  
\[\left( \begin{array}{c}{p_{1}}^{\prime}(i,j,k)\\ \vdots\\ {p_{n}}^{\prime}(i,j,k)\end{array} \right)=\left( \begin{array}{ccc}t_{11}& \cdots&t_{1n}\\ \vdots&\ddots&\vdots\\ t_{n1}&\cdots&t_{nn}\end{array} \right)\cdot\left( \begin{array}{c}p_{1}(i,j,k )\\ \vdots\\ p_{n}(i,j,k)\end{array} \right) \tag{3}\]  
where \(T_{s0\to s1}=\left( \begin{array}{ccc}t_{11}&\cdots&t_{1n}\\ \vdots&\ddots&\vdots\\ t_{n1}&\cdots&t_{nn}\end{array} \right)\) is the state transition matrix, which reflects the transition process of random exit selection of a driver between different states. In the real traffic flow process, not only the probability vector \(P(i,j,k)_{s0}\) and \(P(i,j,k)_{s1}\) but also the transition matrix \(T_{s0\to s1}\) are unobservable. As there are heterogonies in exit selection between different drivers, each driver has a specific transition matrix \(T_{s0\to s1}\). In real-world random exit selection, a driver indeed considers all the feasible choices as a total, which means in the network, he/she chooses every possible node available to generate multiple paths. These paths evolve through time, and the probability distribution of its position at any node is not deterministic. In addition, any changes in the selection of the probability of a node will affect all the probability distribution. As the probability is unobservable, the continuous change of probability will, in turn, create a chain reaction, which will then lead to a combinatorial explosion. This combinatorial explosion makes it difficult, if not impossible, to use deterministic or stochastic methods for both modeling and simulating the random exit selection of a massive number of drivers in expressway traffic. Procedure 3: Mapping the dynamics of probability into real traffic volume. In order to simulate the real volume of traffic flow at each exit of the expressway, it is necessary to resolve the mapping between the dynamic of exit probability and the traffic flow at each exit. Since Procedure 2 only output the probability of whether a driver will choose a certain exit, a calibration is needed to map this probability to the real volume of the traffic flow. Since the observation of the state of each vehicle on the highway is not practical, the calibration and verification of the model can only be operated on the total amount of traffic observed at the exit. Figure 1: The quantum walk expressway traffic simulation model. A generalized calibration model can be constructed as follows:  
\[F_{ikr}=f(P(i,k)) \tag{4}\]  
where \(F_{ikr}\) is the real traffic volume obtained by simulated probability, \(P(i,k)\) is the simulated probability of leaving the expressway from exit \(E_{i}\) at time \(T_{k}\), and \(f\) is the calibration function that maps \(P(i,k)\) to \(F_{ikr}\).