## 3 The RQTM Model
### Map Probability Dynamics to Traffic Volumes  
In the quantum walk-based expressway simulation model, there are two key parameters that need to be determined. One is the time step \(\Delta k\) of the quantum walk evolution, and the other is the correction factor \(a_{i}\) from the probability magnitude to the real traffic data. Assuming that the time series of the traffic flow of the exit obtained by the real observation is \(F_{ik}\), and the quantum walk of the parameter is \(\Delta k\), the \(i\)th-exit probability sequence of the model output is \(P_{i}(k)\), then the quantum walk correction model can be constructed as follows:  
\[F_{ik}=a_{i}P_{i}(k) \tag{17}\]  
In the above model, \(P_{i}(k)\) is a probability determined by the characteristics of the quantum walk. The variable \(k\) determines the characteristics and laws of the quantum walk probability evolution, which determines the time scale and period characteristics of the quantum walk simulation data (see the discussion in Section 5). The parameterization is carried out by means of stepwise approximation. Referring to the determination method of the link coefficient in the classical random walk, we can gradually increase \(k\) with an interval of \(\Delta k\) and compare different simulation results with the measured data in the overall waveform structure and statistics. The best match will produce the optimal \(\Delta k\). After the optimal \(\Delta k\) parameter is determined, the correction coefficient \(a_{i}\) can be obtained by directly fitting the simulation results to the real observation using the least square method.  
As there is a large amount of regular traffic that periodically changes in the traffic flow, which cannot be captured by the quantum walk, we also used the autoregressive moving average model to model the residual series.  
The autoregressive moving average model is composed of the autoregressive model and the moving average model, which is an important method to study time series, and the corresponding parameters are \(p\) and \(q\), respectively. The \(p\) denotes the lag number of the time series data used in the prediction model, expressed as the partial autocorrelation coefficient, which is close to 0 in the \(p\)-order lag, also called the AR/auto regressive term. The \(q\) represents the prediction used in the moving average prediction model. The hysteresis of the error is expressed as the autocorrelation coefficient approaching 0 at the \(q\)-order lag, also known as the MA/moving average term. The overall formula of the autoregressive moving average model is  
\[y_{t}=\mu+\phi_{1}\times y_{k-1}+\cdots+\phi_{p}\times y_{k-p}+\theta_{1} \times\varepsilon_{k-1}+\cdots+\theta_{p}\times\varepsilon_{k-p} \tag{18}\]where \(\phi\) is the coefficient of the auto regressive process, and \(\theta\) is the coefficient of the moving average process.  
To summarize, the overall model of RQTM is  
\[RQTM(F_{ik})=QW(k)+ARMA(p,q)=\alpha_{i}P(i,k)+ARMA(p,q) \tag{19}\]  
where \(i,k\) are the indices of exit and time, respectively, \(\alpha_{i}\) is the correction factor from the probability \(P(i,k)\) to the real traffic data, and \(p\) and \(q\) are the fitting parameter of ARMA.