Quantum harmonic oscillator model for simulation of intercity population mobility

HU Xu; QIAN Lingxin; NIU Xiaoyu; GAO Ming; LUO Wen; YUAN Linwang; YU Zhaoyuan

doi:10.1007/s11442-024-2213-3

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Journal of Geographical Sciences ›› 2024, Vol. 34 ›› Issue (3) : 459-482. DOI: 10.1007/s11442-024-2213-3

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Quantum harmonic oscillator model for simulation of intercity population mobility

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Abstract

The simulation of intercity population mobility helps to deepen the understanding of intercity population mobility and its underlying laws, which has great importance for epidemic prevention and control, social management, and even urban planning. There are many factors that affect intercity population mobility, such as socioeconomic attributes, geographical distance, and industrial structure. The complexity of the coupling among these factors makes it difficult to simulate intercity population mobility. To address this issue, we propose a novel method named the quantum harmonic oscillator model for simulation of intercity population mobility (QHO-IPM). QHO-IPM describes the intercity population mobility as being affected by coupled driving factors that work as a multioscillator-coupled quantum harmonic oscillator system, which is further transformed by the oscillation process of an oscillator, namely, the breaking point of intercity population mobility. The intercity population mobility among seven cities in the Beijing-Tianjin-Hebei region and its surrounding region is taken as an example for verifying the QHO-IPM. The experimental results show that (1) compared with the reference methods (the autoregressive integrated moving average (ARIMA) and long and short-term memory (LSTM) models), the QHO-IPM achieves better simulation performance regarding intercity population mobility in terms of both overall trend and mutation. (2) The simulation error in the QHO-IPM for different-level intercity population mobility is small and stable, which illustrates the weak sensitivity of the QHO-IPM to intercity population mobility under different structures. (3) The discussion regarding the influence degree of different driving factors reveals the significant “one dominant and multiple auxiliary” factor pattern of driving factors on intercity population mobility in the study area. The proposed method has the potential to provide valuable support for understanding intercity population mobility laws and related decision-making on intercity population mobility control.

Key words

intercity population mobility / coupling driving factors / quantum harmonic oscillator model / probability distribution pattern / optimization strategy

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HU Xu, QIAN Lingxin, NIU Xiaoyu, GAO Ming, LUO Wen, YUAN Linwang, YU Zhaoyuan. Quantum harmonic oscillator model for simulation of intercity population mobility[J]. Journal of Geographical Sciences, 2024, 34(3): 459-482 https://doi.org/10.1007/s11442-024-2213-3

1 Introduction

Intercity population mobility is the spatial movement or geographical flow of the population distribution between cities, and it includes both long-term and short-term population transfers (Smolak et al., 2020; Shi et al., 2022). Intercity population mobility is of great importance to urban development: it facilitates social, economic, and cultural exchanges, shapes the form taken by cities, but also gives rise to traffic congestion and pollution, and fuels the spread of infectious diseases such as COVID-19 (Liu, 2018; Tan et al., 2021). Thus, simulating intercity population mobility is not only crucial to clarify the characteristics of intercity population mobility but also to understand the impact of complex factors on intercity population mobility for supporting intercity population mobility control and urban social governance (Chang et al., 2021; Wang et al., 2022).

Driven by various factors, such as socioeconomic attributes, geographical distance, and industrial structure, intercity population mobility drives population reconfiguration among cities. Whether it is the early classical population mobility theory (e.g., the classic “push- pull” theory and its extensions (Lee, 1966)), the population mobility theory of neoclassical economics (e.g., the Dual Sector theory (Lewis, 1954) and the Todaro model (Todaro, 1969)), or research on the impact mechanism of (intercity) population mobility conducted from the perspectives of economy, society, space, and environment in recent years (Shen and Shen, 2020), these theories have all emphasized the fact that various driving factors jointly contribute to intercity population mobility.

Multiple studies have emphasized the essential role and research value of the relevant driving factors by analyzing the relationship between these driving factors and intercity population mobility. The existing intercity population mobility simulation models rarely consider these driving factors. For example, the traditional mechanism-based model is generally used to construct the relationship between intercity population mobility and variables such as geographical distance and population density from the perspective of spatial interaction to simulate intercity population mobility. On the one hand, the parameter calibration of the traditional mechanism-based model is extremely complex. On the other hand, the preset empirical parameters can only be used to reconstruct the overall or average state of intercity population mobility, making it difficult to rely on the traditional mechanism-based model to simulate the finer structure of intercity population mobility, especially during special events and holidays. Another such model is the data-based model, which includes the traditional statistical model and statistical learning model. This model is a mathematical method for considering intercity population mobility and its relevant variables based on hidden information in historical intercity population mobility data. Obviously, the data-based model requires representative high-quality training data to conduct an accurate simulation. Moreover, the data-based model does not account for the influence of relevant endogenous/exogenous variables or even driving factors on intercity population mobility.

However, population mobility, especially intercity population mobility, is a complex nonlinear process of population transfer among cities that is driven by many coupled factors (Wang et al., 2019; Li et al., 2021). For example, the “push-pull” theory suggests that population mobility is the result of the comprehensive effect of intermediate barriers, personal factors, and factors that flow into and out of areas (Lee, 1966). Population theory, which is based on neoclassical economics, also emphasizes the influence of economic factors such as income inequality and employment rate on population mobility (Lewis, 1954; Todaro, 1969). In recent years, the schools of economics and behavior in the field of population mobility have worked to strengthen our understanding of the mechanism underlying the various driving factors on population mobility from the perspectives of economy, society, space, and environment (Sheng, 2018; Shen and Shen, 2020). These driving factors are related and coupled, and they are the essential reasons for intercity population mobility. Simulating intercity population mobility through the consideration of driving factors is not only conducive to developing an in-depth understanding of intercity population mobility and its laws but also helps to clarify the coupling relationship and coupling strength of the underlying driving factors to provide urban managers with reasonable and effective measures for controlling intercity population mobility. As the coupling relationship and coupling strength of driving factors cannot be directly measured, it is difficult to apply the traditional mathematical statistics method to the construction of the intercity population mobility simulation model based on each driving factor.

In this study, the quantum harmonic oscillator model is introduced to simulate intercity population mobility by regarding these driving factors as quantum oscillators. The quantum harmonic oscillator model is an extension of the classical oscillator model (Biamonte et al., 2019). The quantum harmonic oscillator model describes the dynamic evolution of a system under the action of coupled oscillators, and this evolution law can be expressed through the Schrödinger equation (Quesne, 2015). The probability-time distribution that is solved by the Schrödinger equation represents the evolution of oscillator occurrence probability over time (Hsueh et al., 2020), which can be used to describe the evolution of intercity population mobility over time under the coupling effect of time-varying driving factors. Moreover, the application of the quantum harmonic oscillator model in statistics, physics, and other related fields also provides the foundation for its application in intercity population mobility (Linke et al., 2017; Childs et al., 2018). There are two reasons for providing a reasonable and effective connection for applying the quantum harmonic oscillator model to simulate intercity population mobility from the perspective of coupled driving factors. First, in the quantum harmonic oscillator model, the coupled driving factors can be represented by the metaphor of the multioscillator system connected by virtual springs, and the coupling strength of the driving factors can be indicated by the states of these virtual springs (Hsueh et al., 2020). Second, the probability-time distribution solved by the Schrödinger equation represents the proportion of intercity population mobility (Hu et al., 2022a).

This study is aimed at constructing a quantum harmonic oscillator model for the simulation of intercity population mobility, which is abbreviated as QHO-IPM. First, the intercity population mobility affected by the coupled driving factors is abstracted as a multioscillator-coupled quantum harmonic oscillator system, which is further decoupled into a multioscillator-independent quantum harmonic oscillator system. Next, the QHO-IPM is solved by the energy level superposition principle and stereographic projection. Finally, a mapping mechanism is constructed to map the wave function to intercity population mobility. The remainder of this paper is organized as follows: Section 2 summarizes the related works on the driving factors of population mobility, the intercity population mobility simulation model, and quantum computing and its applications. Section 3 introduces the overall framework of the QHO-IPM. The overall methods are detailed in Section 4. In Section 5, a case study is presented to validate the QHO-IPM. The discussions are presented in Section 6, and conclusions are drawn in Section 7.

2 Related works

2.1 Driving factors of population mobility

The driving factors of population mobility have always been one of the key topics in the field of human geography, and they have attracted widespread attention from scholars both domestically and internationally. Early classical theories posit that uneven regional development is the main driving force behind population mobility. “The Laws of Migration” that was proposed in the late 19th century (Ravenstein, 1885), the classic “push-pull” theory (Lee, 1966), and the hypothesis of population mobility transition (Zelinsky, 1971) all emphasize the fact that regional economic development differences serve as the main driving force for population mobility. With the rapid development of neoclassical economics, population theories such as the Dual Sector theory and the Todaro model have emerged under the assumption that population mobility is affected by economic factors such as income inequality and employment rates (Lewis, 1954; Todaro, 1969). Subsequently, some scholars began to criticize the limitations of the above regional imbalance theory, which only emphasizes economic factors. These scholars hold that the influence of factors such as social networks, culture, climate, public facilities, and services is becoming increasingly important, so economic and behavioral schools of population mobility have emerged.

Therefore, many scholars have gradually begun to pay attention to the influence of factors such as economic scale, income level, employment opportunities, industrial structure, administrative division, population density, city scale, geographical distance, transportation connections, public service, and natural environment on population mobility from the perspectives of economy, society, space, and environment (Sheng, 2018; Shen and Shen, 2020). For example, Zhao et al. (2019) revealed the profound influence of economic development indicators on population mobility based on Baidu population mobility data covering the Spring Festival in 2015. Wang et al. (2012) believed that interprovincial population mobility in China is mainly influenced by macro factors such as the regional structure of the geographical environment, livability conditions, and economic differences. The per capita income of inflow areas and the population size of outflow areas are the main driving factors behind population mobility. Tang et al. (2020) used a gravity model to verify that urban characteristics, such as the time cost of population transfer, industrial structure, and cultural education, shape the pattern of intercity population mobility, and urban characteristics have a greater influence on intercity population mobility than time costs. Wang et al. (2020) analyzed the population mobility of the Yangtze River Delta urban agglomeration during the Spring Festival travel rush and daily hours and revealed the driving factors for intercity population mobility, as well as the relationships between these driving factors and intercity population mobility. Throughout the development of the early classical population mobility theory toward greater alignment with the schools of economics and behavior that has occurred in recent years, the academic community has gradually deepened its understanding of the driving factors of population mobility. Moreover, the consensus that population mobility is a spatial movement or geographical flow of population distribution under the comprehensive influence of many driving factors has been reached in the research on the evolution pattern and impact mechanism of population mobility.

2.2 The intercity population mobility simulation model

The models designed for simulating intercity population mobility can be roughly divided into two major categories: traditional mechanism-based models and data-based models. The traditional mechanism-based model simulates intercity population mobility by constructing the relationship between intercity population mobility and the relevant physical variables (e.g., geographical distance, population density, etc.) based on the physical mechanism or mathematical equation abstracted from intercity population mobility (Barbosa et al., 2018; Toch et al., 2019), such as the gravity model (Zipf, 1946), intervening opportunity model (Stouffer, 1940), radiation model (Simini et al., 2012), field model (Mazzoli et al., 2019), population weighted opportunity (PWO) model (Yan et al., 2014), and their extension models (Yan et al., 2017; Liu and Yan, 2019; Simini et al., 2021). The gravity model is a power rate model based on the assumption that the population mobility between two cities is a decreasing function of their geographical distance. The gravity model and its improved extension models are rough simplifications of intercity population mobility, but they are very sensitive to fluctuations in intercity population mobility (Simini et al., 2021). Moreover, for modern society with its extremely developed forms of transportation, the assumption that the geographical distance between two cities is the main factor affecting intercity population mobility is gradually being invalidated. The intervention opportunity model holds that in addition to the city scale and the geographical distance between the outflow city and the inflow city, the possible “intervention opportunities” between the two cities are also a key to intercity population mobility. The disadvantage of this model is that it is complex and has poor simulation and prediction performance for intercity population mobility between close cities (Liu and Yan, 2020). For situations where detailed population mobility data cannot be obtained, Simini et al. (2012) proposed the radiation model, which is a spatial distribution model for simulating population mobility that does not require calibration with observation data. This model simplifies parameter adjustment operations in modeling but weakens the description of some real intercity population mobility. In addition, some scholars have advanced a field model, which is used to abstract intercity population mobility as a vector field for describing fluid flow and applies theories such as the Gaussian divergence theorem to characterize the integrity and complexity of intercity population mobility.

Based on the above models, scholars proposed the PWO model (Yan et al., 2014), the universal opportunity (UO) model (Yan et al., 2017), and the opportunity priority selection (OPS) model (Liu and Yan, 2019). These emerging models account for both group and individual mobility patterns, transcending the classical gravity law. Most of the above models simulate (intercity) population mobility from the perspective of spatial interaction, revealing the evolution laws, spatiotemporal differentiation, and spatial dependence of (intercity) population mobility. However, these models involve complex parameter calibration, which makes it difficult to scientifically calibrate the parameters across different research objects, spatiotemporal backgrounds, and model settings. Even if some studies adopt preset empirical parameters, these models can reproduce only the overall or average state of intercity population mobility but cannot simulate the finer structures of intercity population mobility, especially when intercity population mobility is affected by special events or holidays.

Another popular data-based model, which includes the traditional statistical model and statistical learning model, has been applied to population mobility prediction (Xie et al., 2020), traffic flow estimation (Avila and Mezić, 2020), and COVID-19 prediction (Zhao et al., 2022). Usually, the traditional statistical model regards intercity population mobility as a random process and then extracts or learns structural patterns from large amounts of intercity population mobility data to simulate intercity population mobility. Due to its simplicity and high computational efficiency, the autoregressive integrated moving average (ARIMA) is widely used in simulation and prediction research on time series data such as population mobility (Ceylan, 2020). However, such models require that the population mobility data is kept at an equilibrium state or satisfies the stationarity assumption. Due to the nonstationary intercity population mobility time series, especially those affected by abnormal events or holidays, the simulation results of the traditional statistical model may have large deviations. The statistical learning model trains or infers a “black box” model to simulate intercity population mobility by continuously adjusting the parameters for processing a large amount of historical data. Time-based statistical learning models (e.g., the recurrent neural networks (RNNs), long and short-term memory (LSTM), and their extended models) and space-based statistical learning models (e.g., the convolutional neural networks (CNNs), the graph convolutional networks (GCNs), and their extended models) are typical statistical learning models (Fu et al., 2018; Zeroual et al., 2020). A statistical learning model can capture the spatiotemporal correlation characteristics of intercity population mobility and has the ability to identify the complexity and nonlinear characteristics of intercity population mobility (Zhang et al., 2018). However, the simulation performance of the statistical learning model is strongly dependent on the quality of the training data. Moreover, the overfitting, complex parameters, and difficulty of explanation also limit their applicability to the simulation of large-scale long-term intercity population mobility (Yuan et al., 2021b). In summary, both the traditional statistical and statistical learning models are data-driven simulation models for intercity population mobility that do not consider the influence of relevant endogenous/exogenous variables or even driving factors on intercity population mobility.

In addition, existing studies on population mobility simulation have mostly focused on intracity population mobility, with relatively little attention given to intercity population mobility. At the same time, existing studies on the modeling and simulation of intercity population mobility have mostly focused on short-term population mobility during leisure travel and home visits during holidays (e.g., the Spring Festival travel rush and the National Day) (Zhao et al., 2017; Zhao et al., 2019), while relevant studies on long-term intercity population mobility, including holidays and nonholidays, are relatively lacking. The multidimensional composite superposition of factors such as intercity business connections, capital flows, and leisure tourism reflected by this long-term intercity population mobility is of great significance for exploring intercity connections and regional spatial organization. Therefore, it is necessary to model and simulate long-term intercity population mobility.

2.3 Quantum computing and its applications

Benefiting from quantum properties such as superposition and entanglement, quantum computing transcends the limitations of traditional computing (Nimbe et al., 2021). Since Deutsch proposed the first quantum algorithm in 1985 (Deutsch, 1985), many scholars have proposed various quantum algorithms, including the large number decomposition algorithm (Shor, 1994), quantum search algorithm (Grover, 1997), quantum random walk algorithm (Kempe, 2003), large linear equations solving algorithm (Harrow et al., 2009), big data least squares fitting algorithm (Wiebe et al., 2012), support vector machine quantum algorithm (Rebentrost et al., 2014), variational quantum eigenvalue solving algorithm (Peruzzo et al., 2014), and quantum approximation optimization algorithm (Medvidović and Carleo, 2021). In recent years, advanced quantum algorithms, such as neural network quantum algorithms and quantum machine learning in the fields of intelligent computing (IC), machine learning (ML), and deep learning (DL) (Chalumuri et al., 2020; Cerezo et al., 2022) have been developed, which has solved some classic computational difficulties (Daley et al., 2022).

In the field of eco-environmental surface modeling, Yue et al. (2022, 2023) creatively combined high accuracy surface modeling (HASM) with the Harrow-Hassidim-Lloyd (HHL) algorithm to form the HASM-HHL quantum machine learning algorithm. HASM-HHL has been successfully applied in the fields of spatial interpolation, spatial upscaling/downscaling, data fusion, and model-data assimilation of eco-environment surfaces. In addition, HASM- HHL provides a general framework for applications such as constructing digital terrain models, simulating climate change, estimating carbon stocks and CO₂ concentrations, modeling soil properties, developing COVID-19 scenarios, computing biodiversity dynamics, and analyzing ecosystem responses to climate change. In addition, quantum computing has also demonstrated superiority in addressing specific problems in fields such as traffic computing (simulation and prediction (Yu et al., 2022), feature analysis (Hu et al., 2022b), traffic optimization (Hussain et al., 2020), traffic control (Xiao et al., 2020)), big data retrieval, financial services, artificial intelligence, biochemistry, and aerospace (Dunjko and Briegel, 2018; Gill et al., 2022). The rapid development and widespread application of quantum computing have laid a feasible foundation for the simulation of intercity population mobility. However, there are few empirical studies on population mobility that are based on the quantum harmonic oscillator model. The multilevel and superposition states of the quantum harmonic oscillator model make it widely used in algorithm design, intelligent optimization, spatial search, and even data analysis. Therefore, the quantum harmonic oscillator model may have the potential to model, simulate, and analyze large-scale intercity population mobility scenarios.

3 Basic concept

Taking the one-way intercity population mobility

A \to B

B \to C

, and

C \to A

among cities A, B, and C as an example. It is assumed that these three types of one-way intercity population mobility operate independently and do not affect each other and that they are formed under the coupling effect of many driving factors. The overall framework of QHO-IPM is shown in Figure 1.

Figure 1 The overall framework of QHO-IPM

Full size|PPT slide

For intercity population mobility

A \to B

, it is assumed that this intercity population mobility is only influenced by two driving factors: economic scale (ES) and income level (IL). The ES of city A is smaller than that of city B, but its IL is higher than that of city B. According to the classic “push-pull” theory (Lee, 1966), intercity population mobility results from a comprehensive effect of factors that flow into and out of cities. As a reference, the two driving factors, ES and IL, both produce opposite forces on intercity population mobility, forming the classic oscillator system of intercity population mobility, as shown in the intercity population mobility

A \to B

in Figure 1a. The red arrow indicates a larger driving force, and the blue arrow represents the opposite. The attractiveness of ES and IL to the population in city B is greater than that in city A, so the population transfers from city A to city B under the comprehensive effect of ES and IL.

However, the influence of driving factors on intercity population mobility is time-varying, especially when affected by abnormal events or holidays. For example, population density and geographical distance may be the dominant driving factors for intercity population mobility during nonholiday periods, while that during holidays such as National Day may be leisure tourism. Unfortunately, the time-varying characteristics of the driving factors are not yet clear, which limits the ability of the classical oscillator system with fixed parameters to abstract the time-varying driving factors and their driving forces. That is, it is almost impossible for a classical oscillator system to characterize the evolution law between driving factors and intercity population mobility. Therefore, it is necessary to further transform and upgrade the classical oscillator system.

How to consider time-varying driving factors in an elegant way for modelling and simulating intercity population mobility is the key problem in this study. Drawing on the uncertainty of quantum theory, the classical oscillator system can be transformed into a quantum oscillator system to realize the consideration of time-varying driving factors. Then, a quantum harmonic oscillator model for the simulation of intercity population mobility can be constructed accordingly, as shown in the intercity population mobility

B \to C

in Figure 1a. In QHO-IPM, the time-varying driving factors are abstracted as quantum oscillators, and the probability-time distribution of the system under the coupling of these quantum oscillators characterizes the time-varying state of intercity population mobility. Specifically, when the driving factors and their driving forces are constant, the virtual spring is in an equilibrium state, and the intercity population mobility is relatively stable. When the driving factors and their driving forces change, the probability-time distribution of the system changes with the stretching/compression of the virtual springs, and therefore, the intercity population mobility changes. In addition, the virtual springs connecting quantum oscillators and their state/elasticity coefficients represent the coupling relationship and coupling strength between driving factors, respectively.

Another difficulty faced by this study is solving the QHO-IPM. The coupling problem of the harmonic oscillator is a classical problem in quantum mechanics, solid physics, etc. (Park, 2019). Thus, the coordinate transformation is applied to decouple the coupled quantum harmonic oscillator system into a multioscillator-independent quantum harmonic oscillator system. Unfortunately, the unclear time-varying characteristics of the driving factors limit the solution of the wave function of the independent oscillators, making it impossible to obtain the system wave function. A feasible approach is simplifying the multioscillator-independent quantum harmonic oscillator system into a single oscillator quantum harmonic oscillator model, and then the system wave function can be described by the wave function of several discrete energy levels of the single harmonic oscillator (Bonezzi et al., 2017). That is, intercity population mobility can be described as the oscillation process of an oscillator connected to the inflow city, namely, the breaking point of intercity population mobility, as shown in Figure 1b. This breaking point continuously oscillates near the equilibrium position, which can approximate the dynamic evolution of intercity population mobility. The oscillation amplitude of the breaking point is used to quantify the scale of intercity population mobility.

The energy level superposition principle provides an idea for solving the above quantum harmonic oscillator model with a single oscillator. The quantum harmonic oscillator model can be solved as a probability-time distribution superimposed with probabilities of different energy levels, which represents the proportion of intercity population mobility under different energy levels to the total intercity population mobility. The influence degree of driving factors can be further estimated, as shown in Figure 1c. Finally, the mapping mechanism between this probability-time distribution and intercity population mobility is established, and the parameter estimation and mapping simulation are realized through the assistance of the observed intercity population mobility data, as shown in Figure 1d.

4 Methodology

In this study, the construction of a quantum harmonic oscillator model for intercity population mobility is attempted to realize the simulation of intercity population mobility affected by coupled driving factors. QHO-IPM includes the following four steps. First, a multioscillator-coupled quantum harmonic oscillator system is constructed by considering the driving factors of intercity population mobility. Next, the decoupling strategy of generalized coordinate transformation is adopted to simplify the above coupled quantum harmonic oscillator system into a quantum harmonic oscillator model. Then, the wave function of the quantum harmonic oscillator model is solved according to the energy level superposition principle and a stereographic projection. Finally, the mapping mechanism between the probability-time distribution and intercity population mobility is constructed to realize the reconstruction and simulation of intercity population mobility.

4.1 The coupled quantum oscillator system

This study regards each driving factor as an oscillator and constructs a multioscillator-coupled quantum harmonic oscillator system. In this system, the driving factors are denoted as m_i. According to quantum theory, the Hamiltonian of the above coupled quantum harmonic oscillator system is as follows (Park, 2019):

\hat{H} = \sum_{i = 1}^{n} a_{i} m_{i}^{2} + 2 \sum_{i, j = 1}^{n} c_{i j} m_{i} m_{j}

(1)

where the quadratic form of the coupled quantum harmonic oscillator system characterizes the intercity population mobility process between cities under the coupling of driving factors m_i and m_j. Here, a_i is the coefficient of the uncoupled term, and c_ij is the coupling coefficient between the driving factors m_i and m_j.

Nevertheless, determining how to solve the coupling problem of coupled quantum harmonic oscillator systems is a common problem in some fields (e.g., quantum theory and solid-state physics), so Equation (1) cannot be solved directly. One possible solution is to seek a decoupling strategy to transform the coupled quantum oscillator system into a multioscillator-independent quantum harmonic oscillator system and further construct the quantum harmonic oscillator model to determine the breaking point for intercity population mobility.

4.2 Decoupling the coupled quantum oscillator system

In this study, the generalized coordinate transformation is applied to decouple the above coupled quantum harmonic oscillator system. Under the new base, the Hamiltonian of the coupled quantum harmonic oscillator system is re-expressed in the standard quadratic form to eliminate the coupling term. Then, Equation (1) can be expressed in matrix form as:

\hat{H} = [m_{1}, \dots, m_{n}] [\begin{array}{ccccc} a_{1} & c_{12} & c_{13} & \dots & c_{1 n} \\ c_{21} & a_{2} & c_{23} & \dots & c_{2 n} \\ c_{31} & c_{32} & a_{3} & \dots & c_{3 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ c_{n 1} & c_{n 2} & c_{n 3} & \dots & a_{n} \end{array}] [\begin{matrix} m_{1} \\ ⋮ \\ m_{n} \end{matrix}] = M^{T} B M

(2)

where M^T=[m₁, m₂, …, m_n]. The coupling matrix B is transformed into a diagonal matrix according to quadratic theory. Then, the coupling term in Equation (1) can be eliminated:

(m_{i}^{'}) = S^{T} (m_{i})

(3)

where S, the unitary transformation matrix, comprises eigenvalues {λ₁, λ₂, …, λ_n} and the corresponding eigenvectors {A₁, A₂, …, A_n} of the matrix B, can be recorded as (Zúñiga et al., 2017):

S^{T} = {[A_{1}, A_{2}, \dots, A_{n}]}^{T}

(4)

Furthermore, the Hamiltonian

\hat{H}

of the coupled quantum harmonic oscillator system based on the above transformation can be converted into the following form:

\hat{H} = M^{T} B M = M^{' T} {(S^{- 1})}^{T} B S^{- 1} M^{'}

(5)

where

{(S^{- 1})}^{T} B S^{- 1} = [\begin{array}{lllll} λ_{1} \\ λ_{2} \\ λ_{3} \\ ⋱ \\ λ_{n} \end{array}]

(6)

After decoupling through the above matrix transformation, the Hamiltonian can be further transformed into:

\hat{H} = λ_{1} m_{1}^{' 2} + λ_{2} m_{2}^{' 2} + \dots + λ_{n} m_{n}^{' 2}

(7)

Our goal is to solve the wave functions of the above decoupled quantum harmonic oscillator system, namely, the multioscillator-independent quantum harmonic oscillator system. The system wave function

ψ (x)

can be expressed as the tensor product of the wave functions

{ψ_{1} (x), ψ_{2} (x), \dots, ψ_{n} (x)}

of each harmonic oscillator in the multioscillator-independent quantum harmonic oscillator system:

ψ (x) = ψ_{1} (x) \otimes ψ_{2} (x) \otimes \dots \otimes ψ_{n} (x)

(8)

However, the number of driving factors for intercity population mobility is unknown, which makes it almost impossible to solve the above system wave function,

ψ (x)

. At this point, further determining how to solve the abovementioned wave functions

{ψ_{1} (x), ψ_{2} (x), \dots, ψ_{n} (x)}

efficiently is needed.

4.3 Optimization and solution of the quantum harmonic oscillator model

Inspired by the energy level superposition principle, the multioscillator-independent quantum harmonic oscillator system is approximated as a quantum harmonic oscillator model with a single oscillator, i.e., the quantum harmonic oscillator model of the breaking point for intercity population mobility. Furthermore, the system wave function

ψ (x)

can be approximated as follows:

ψ (x) = \sum_{k = 0}^{K} ξ_{k} ψ_{k} (x)

(9)

where

ψ_{0} (x), ψ_{1} (x), \dots, ψ_{K} (x)

are the wave functions of the 0th (the ground state), 1st, …, kth energy levels, respectively. In quantum theory,

ξ_{k}

indicates the probability amplitude of the wave function at the kth energy level. Through the Taylor series expansion of the wave function, the wave function of the kth energy level can be expressed as (Grining et al., 2015):

ψ_{k} (x) = {(\frac{1}{\sqrt{π} 2^{k} k!})}^{1 / 2} \cdot e^{- x^{2} / 2} \cdot H_{k} (x)

(10)

And the Hamiltonian basis can be defined as

h_{k} (x) = (e^{- x^{2} / 2} / {(\sqrt{π} 2^{k} k!)}^{1 / 2}) \cdot H_{k} (x)

According to quantum theory, the probability density function f(x) is the square of the wave function

ψ (x)

f (x) = | ψ (x) |^{2} = {(\sum_{k = 0}^{K} ξ_{k} h_{k} (x))}^{2}

(11)

where

\sum_{k = 0}^{K} ξ_{k}^{2} = 1

. In QHO-IPM,

ξ_{k}

represents the influence degree of the kth type driving factor on intercity population mobility. The distribution of

ξ_{k}

can reflect the differences in the driving factors of intercity population mobility among different cities, it is an important basis with which urban managers and planners can formulate reasonable intercity population mobility control measures. Therefore, we go on to discuss the spatial and temporal distribution of

ξ_{k}

to clarify the spatiotemporal characteristics of the driving factors among different cities.

In the quantum harmonic oscillator model, parameter optimization is another problem to be solved. In QHO-IPM, the energy level coefficient

ξ_{k}

needs to be solved to further study the characteristics of the driving factors among different cities. Given that

\sum_{k = 0}^{K} ξ_{k}^{2} = 1

and the sum of the integrals of probability density f(x) is 1, a spherical model is constructed accordingly. The radius of the spherical model is 1, and any point

(ξ_{1}, ξ_{2}, \dots, ξ_{K})

appears on the sphere. Therefore, a stereographic projection transformation is performed in this study: any point on the sphere is projected onto the two-dimensional plane

Π

through the origin. Hence,

ξ_{k}

on the sphere is converted to

ϑ_{k}

on a plane. Based on the projected area

S = \sum_{k = 0}^{K} ϑ_{k}^{2}

, the probability form can be expressed as:

P (ξ_{k}) = v_{k} / S

(12)

Furthermore, based on the optimal

ϑ_{k}

, the reversed stereographic projection is adopted to optimize

ξ_{k}

P^{- 1} (ϑ_{k}) = {\begin{cases} 1 - 2 / (S + 1), if k = 0 \\ 2 ϑ_{k} / (S + 1), if 1 ⩽ k ⩽ K \end{cases}

(13)

Therefore, the log likelihood at

{x_{1}, x_{2}, \dots, x_{I}}

is:

ℓ (ϑ) = \sum_{i = 1}^{I} \log {(\sum_{k = 0}^{K} {[P^{- 1} (ϑ)]}_{k} \cdot h_{k} (x_{i}))}^{2}

(14)

Obviously, the monotonic likelihood function

ℓ (ϑ)

in Equation (14) can be solved using common optimization methods. Finally, the system wave function is solved.

4.4 Mapping time-varying probability to population mobility

The purpose of the intercity population mobility simulation conducted in this study is to simulate the time series of the intercity population mobility between two cities. In QHO-IPM, the oscillation amplitude, i.e., x, of the quantum harmonic oscillator model reflects the scale of intercity population mobility at different times. Then, the probability density f(x) can be further transformed into the time-varying probability f(t), which is proportional to the observed intercity population mobility. Therefore, the mapping mechanism between the time-varying probability f_s(t) and intercity population mobility time series F_s(t) can be expressed as follows:

{\begin{matrix} F_{1} (t) = α_{1} f_{1} (t) + β_{1} \\ F_{2} (t) = α_{2} f_{2} (t) + β_{2} \\ ⋮ \\ F_{s} (t) = α_{s} f_{s} (t) + β_{s} \end{matrix}

(15)

where α_s(β_s) is the mapping coefficient, which represents the proportion (number) of intercity population mobility that can (cannot) be captured by f_s(t). The least squares method can be applied to estimate the mapping coefficients α_s and β_s, thus enabling the intercity population mobility to be simulated.

5 Implementation and results

5.1 Data description and experimental configuration

The Beijing-Tianjin-Hebei region is the economic core area of northern China and spans the two municipalities of Beijing and Tianjin as well as Hebei Province (Fan et al., 2022; Yang and Fan, 2022). As one of the three major urban agglomerations in China, the intercity population mobility in the Beijing-Tianjin-Hebei region exhibits the characteristics of multiple patterns and fast frequency, which may characterize the intercity population mobility in most regions of China (Yuan et al., 2021a; Huang et al., 2022).

In this study, seven cities at different levels, namely, five cities in the Beijing-Tianjin- Hebei region (i.e., Tianjin, Langfang, Baoding, Cangzhou, and Hengshui) and two surrounding cities (i.e., Dezhou and Liaocheng), are selected as the study area, as shown in Figure 2a. The daily intercity population mobility data of the intercity population mobility network, including Tianjin-Langfang, Langfang-Baoding, Langfang-Cangzhou, Baoding- Cangzhou, Cangzhou-Hengshui, Hengshui-Dezhou, and Dezhou-Liaocheng, are collected from the Tencent location big data platform (https://heat.qq.com/). The time range runs from January 1, 2016, to December 31, 2018 (1096 days in total). Furthermore, according to the “2023 China Mobile Internet Half Year Report” (https://www.questmobile.com.cn/research/report/1686624886410285058), as of June 2023, the number of Tencent APP users among mobile internet users reached 1.213 billion. Tencent population mobility data have the advantages of a large number of users and a high spatiotemporal resolution. These data have been widely used in research on intercity population mobility, urban agglomeration identification, and intercity travel networks, and their effectiveness has been verified (Pan and Lai, 2019). The average daily intercity population mobility in the study area is shown in Figure 2b. Obviously, the above intercity population mobility among different-level cities gradually increases from the south to the north of the study area, showing significant spatial heterogeneity, which may demonstrate the sensitivity and generalizability of QHO-IPM for modeling and simulating intercity population mobility at different scales.

Figure 2 Study area and average intercity population mobility

Full size|PPT slide

Based on the above intercity population mobility data, the following experiments and data analysis are performed: (1) The first is a QHO-IPM simulation experiment applied to model and simulate intercity population mobility in the above intercity population mobility network. (2) The next is a simulation performance comparison of different methods. The autoregressive integrated moving average (ARIMA) model and the long and short-term memory (LSTM) networks are selected as reference methods, and they are applied to the same intercity population mobility data. The simulation performances of QHO-IPM, ARIMA, and LSTM are further analyzed and compared. (3) The third is an error analysis of the different methods. The error distribution characteristics of the different methods are analyzed, and the modeling characteristics and application of different methods are explained accordingly. In addition to these experiments and analysis, the following discussion is supplemented: The distribution of the influence degree of driving factors is further discussed using the monthly influence degree of three driving factors on intercity population mobility. In addition, the possible improvements of QHO-IPM are also discussed.

Specifically, for the simulation and performance comparison experiments, i.e., experiments (1) and (2), the daily intercity population mobility data from 2016 to 2018 are applied to fit the QHO-IPM. For experiment (1), a 3-order quantum harmonic oscillator model is selected to model and simulate the intercity population mobility. For experiment (2), the d-order difference is applied to transfer the daily intercity population mobility data into a stationary time series. Then, the parameters p and q are obtained by analyzing the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the transformed stationary time series. The optimal ARIMA(p,d,q) is finally determined by applying the Bayesian Information Criterion (BIC). In addition, the optimal network structure of LSTM is a two-layer network structure, including four neurons in the hidden layer and one neuron in the output layer. Adam is used as the optimizer, and the mean square error (MSE) is selected as the loss function. The epochs and batch size are 100 and 30, respectively.

Three criteria are selected to evaluate the simulation performance of the QHO-IPM, namely, the mean absolute error (MAE), root mean square error (RMSE), and coefficient of determination (R²). The MAE, RMSE, and R² are defined in Table 1. In Table 1,

M_{i}

{\bar{M}}_{i}

, and

{\hat{M}}_{i}

are used to indicate the observed intercity population mobility, the mean observed intercity population mobility, and the simulated intercity population mobility, respectively. m is the time series length of intercity population mobility.

Table 1 Definition of evaluation criteria

Evaluation criteria	Definition
Mean absolute error (MAE)	$M A E = \frac{1}{m} \sum_{i = 1}^{m} \| M_{i} - {\hat{M}}_{i} \|$
Root mean square error (RMSE)	$R M S E = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {(M_{i} - {\hat{M}}_{i})}^{2}}$
Coefficient of determination (R²)	$R^{2} = 1 - \sum_{i = 1}^{m} {\| M_{i} - {\hat{M}}_{i} \|}^{2} / \sum_{i = 1}^{m} {\| M_{i} - {\bar{M}}_{i} \|}^{2}$

5.2 Simulation results of intercity population mobility

For the simulation of intercity population mobility, the application of different methods should simulate the average state of intercity population mobility. In addition, whether the intercity population mobility caused by abnormal events or holidays can be accurately simulated is another important indicator when testing the performance of the method.

For two typical cities, Langfang and Baoding, the simulated intercity population mobility of QHO-IPM is compared with that of ARIMA and LSTM, respectively, as shown in Figure 3. In Figure 3, the lines with different colors represent the observed intercity population mobility and the simulation results of different methods. The orange part indicates the intercity population mobility mutation caused by holidays. These holidays are New Year's Day, Qingming Festival, Dragon Boat Festival, Labor Day, National Day, Mid-Autumn Festival, and Spring Festival. On the long-term scale, these three methods can be used to roughly simulate the overall trend of intercity population mobility and reveal its fluctuation law over time. With the assistance of the 1:1 diagram between the simulated intercity population mobility of different methods and the observed intercity population mobility (Figure 4), it is clear that while the simulation results of the reference methods are concentrated on both sides of the 1:1 line, the simulation results of QHO-IPM, especially those points with larger intercity population mobility, are generally closer together and concentrated on the 1:1 line. Therefore, the simulation effect of QHO-IPM on the overall trend and law of intercity population mobility is better than that of the reference methods. This is because QHO-IPM models intercity population mobility from the perspective of driving factors, which reflects the large-scale structure and overall change characteristics of intercity population mobility from the essential mechanisms of intercity population mobility.

Figure 3 The simulated intercity population mobility using different methods

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Figure 4 The 1:1 diagram between the simulated intercity population mobility and the observed intercity population mobility

Full size|PPT slide

In addition, the simulated intercity population mobility of QHO-IPM, ARIMA, and LSTM show significant differences on the small time scale. In particular, QHO-IPM can capture the intercity population mobility mutation caused by holidays, as shown by the orange shading in Figure 3. Taking the National Day in 2016, Labor Day in 2017, and New Year's Day in 2018 as examples, as shown in Figures 3a, 3b and 3g, it is obvious that the intercity population mobility simulated by QHO-IPM has almost the same trend and extreme value as the observed intercity population mobility, while the simulation results of the reference methods for intercity population mobility on holidays are relatively worse. QHO-IPM can also simulate small peaks well, as shown in Figures 3c-3h. In these six attached figures, QHO-IPM accurately simulates all high-frequency oscillations and sudden changes. Further, ARIMA and LSTM are slightly less capable of capturing high-frequency oscillations and mutations, especially irregular small peaks. QHO-IPM, the mechanism model for intercity population mobility, can be used to accurately reveal the relationship between intercity population mobility and the driving factors, even under abnormal situations. However, ARIMA and LSTM are highly dependent on the quantity and quality of training data. Both need to learn the structures and patterns through these data. In the case of a small amount of intercity population mobility mutation data, insufficient learning, and even structural errors, generally result. Therefore, ARIMA and LSTM are slightly worse than QHO-IPM in simulating the intercity population mobility mutation caused by abnormal events, especially that caused by holidays.

The evaluation indicators of the different methods are shown in Table 2. For all intercity population mobility assessed in this study, QHO-IPM shows a higher R² and lower MAE and RMSE. This result illustrates the strong ability of QHO-IPM to capture the irregular fluctuations of intercity population mobility. The simulation accuracy (R²) of the intercity population mobility of Dezhou-Liaocheng is 0.9108, which is the highest among all considered instances of intercity population mobility. For the simulated intercity population mobility of Dezhou-Liaocheng, compared with that of ARIMA and LSTM, the MAE of QHO- IPM is decreased by 28.4% and 14.5%, the RMSE of QHO-IPM is reduced by 32.9% and 21.5%, and the R² of QHO-IPM is improved by 13.5% and 7.8%, respectively. For all intercity population mobility, compared with ARIMA and LSTM, the average MAE of QHO-IPM is decreased by 30.0% and 5.1%, the average RMSE is reduced by 42.7% and 15.8%, and the average simulation accuracy (R²) is increased by 16.7% and 7.6%, respectively. The above analysis also demonstrates that QHO-IPM shows remarkable simulation performance compared with the reference methods.

Table 2 The evaluation indicators of different methods

Origin-Destination	QHO-IPM			ARIMA			LSTM
Origin-Destination	MAE	RMSE	R²	MAE	RMSE	R²	MAE	RMSE	R²
Tianjin-Langfang	2150.11	3078.72	0.8385 (↑)	3277.05	4682.17	0.6265	2116.94	3380.70	0.7943
Langfang-Tianjin	1958.65	2633.53	0.8804 (↑)	2204.80	3588.58	0.7780	1958.15	3182.78	0.8124
Langfang-Baoding	1423.74	1996.32	0.8046 (↑)	2140.35	3164.85	0.5089	1442.95	2358.47	0.7306
Baoding-Langfang	1402.61	1869.96	0.8159 (↑)	1901.29	2849.91	0.5723	1506.91	2400.10	0.6993
Langfang-Cangzhou	919.85	1291.55	0.8701 (↑)	1223.93	1888.65	0.7223	1062.79	1603.53	0.7833
Cangzhou-Langfang	865.24	1195.73	0.8859 (↑)	1043.80	1706.09	0.7676	876.13	1476.18	0.8069
Baoding-Cangzhou	1105.96	1624.13	0.8978 (↑)	1485.26	2425.42	0.7722	1230.12	2040.40	0.8170
Cangzhou-Baoding	1074.19	1544.63	0.9036 (↑)	1340.90	2197.49	0.8048	1167.37	1910.61	0.8303
Cangzhou-Hengshui	810.82	1147.95	0.9009 (↑)	1035.55	1551.71	0.8189	972.41	1413.72	0.8404
Hengshui-Cangzhou	872.09	1209.49	0.8845 (↑)	1005.11	1525.32	0.8163	937.35	1359.53	0.8452
Hengshui-Dezhou	623.81	912.34	0.8960 (↑)	663.17	1088.31	0.8520	591.43	930.64	0.8837
Dezhou-Hengshui	604.51	956.79	0.8947 (↑)	612.57	1040.43	0.8755	620.31	957.81	0.8863
Dezhou-Liaocheng	604.81	932.21	0.9108 (↑)	844.78	1388.26	0.8021	707.25	1187.19	0.8449
Liaocheng-Dezhou	628.42	932.13	0.9107 (↑)	787.05	1338.48	0.8158	665.63	1137.78	0.8565
Mean	1074.63	1523.25	0.8782 (↑)	1397.54	2173.98	0.7524	1132.55	1809.96	0.8165

Note: ↑(↓) indicates that the simulation accuracy (R²) of QHO-IPM is superior to (inferior to) that of the reference methods.

5.3 Error analysis

The error distribution of the different methods considered is shown in Figure 5. Obviously, the simulation error of the QHO-IPM is small, and most of such errors are concentrated at approximately 0. It can be seen from the violin plot and the box plot that the simulation error distribution range of ARIMA is the widest among these three methods. The reason for this is that ARIMA is a traditional statistical method that usually learns structures and patterns for simulation and prediction from historical intercity population mobility data. However, limited training data may result in ARIMA's insufficient learning of the above structures and patterns, especially in regard to mutations caused by abnormal events, making it difficult to accurately reconstruct the fluctuations of intercity population mobility. Moreover, LSTM usually adjusts its parameters at the intersection of peaks and troughs, which causes the simulation effect to be slightly worse. That is, ARIMA and LSTM can almost only simulate the average state of intercity population mobility but cannot accurately respond to the intercity population mobility mutations that are caused by abnormal events. Therefore, the simulation error of QHO-IPM is evenly distributed at approximately 0, and its maximum error is smaller than that of ARIMA and LSTM. The error analysis of the simulation results of the three methods further illustrates the strong simulation ability of QHO-IPM for intercity population mobility.

Figure 5 Error distribution of the different methods

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6 Discussion

6.1 Influence degree of the different factors

The QHO-IPM is further applied to explore the monthly influence degree of the three factors considered in this study on intercity population mobility in the study area, as shown in Figure 6. Figures 6a, 6b, and 6c are the monthly influence degrees of factor 1, factor 2, and factor 3 on intercity population mobility, respectively. The color ranging from blue to red indicates the influence degree ranging from weak to strong.

Figure 6 Distribution of the monthly influence degree of different factors

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The influence degree of factor 1 (Figure 6a) is 0.8305-0.9928, which shows the leading role of this factor in intercity population mobility. On an annual scale, the influence degree of factor 1 on intercity population mobility is relatively stable in 2016, while its influence degree changed slightly in 2017 and 2018 (the standard deviations of the influence degree of factor 1 in 2016, 2017, and 2018 are 0.024, 0.026, and 0.029, respectively). This is because the traffic integration layout of roads and railways in the Beijing-Tianjin-Hebei region was gradually formed in 2016. Travel in the study area is less limited by objective factors, such as time and location constraints, but is more likely to be influenced by individual subjective factors. Therefore, the traffic integration layout in the Beijing-Tianjin-Hebei region makes the influence of factor 1 on intercity population mobility relatively stable in 2016. In particular, in March 2017, the influence of factor 1 on the intercity population mobility of Baoding-Langfang was relatively small (0.8306). This is because the influence of policy factors on intercity population mobility increased after the implementation of the Langfang household registration policy in January 2017, so the influence of factor 1 weakened over a short time. In August 2018, factor 1 had a significant influence on all intercity population mobility in the study area, verifying that the “one-hour life circle” in the Beijing-Tianjin- Hebei region is gradually making some achievements. In addition, the change in the monthly influence degree of factor 1 demonstrates that under the dominance of this factor, intercity population mobility is also affected by other factors. That is, intercity population mobility is a dynamic and nonlinear process that is affected by many driving factors.

Furthermore, factor 2 (Figure 6b) and factor 3 (Figure 6c) are the secondary driving factors of intercity population mobility in the study area. Compared with the other two monthly influence degrees, factor 2 was particularly significant in April 2016, especially for Baoding-Langfang. Affected by major holidays, such as New Year's Day, Spring Festival, and winter and summer vacations, factor 2 had little influence on intercity population mobility at the beginning, middle, and end of the year, so the influence of factor 2 is low around January, July, and December of every year. Factor 3, aside from its weak influence in August 2018, is relatively stable in other periods. This may be because August 2018 marked the end of the summer vacation, the peak tourism season in the Beijing-Tianjin-Hebei region, and the opening season for many schools. At the same time, the operation of tourist trains in the Beijing-Tianjin-Hebei region in August 2018 affected people's travel, resulting in the frequency and pattern of intercity population mobility differing from those in other periods, so the influence of factor 3 in August 2018 is abnormal.

Overall, the average influence degree of factor 1 (0.9455) is significantly higher than that of factor 2 (0.0077) and factor 3 (0.0468). This result demonstrates that (1) factor 1 is the main driving factor of intercity population mobility in the study area and (2) intercity population mobility is mostly dominated by a single driving factor and assisted by many other driving factors. Therefore, a significant “one dominant and multiple auxiliary” factor pattern of the driving factors of intercity population mobility is gradually taking shape.

6.2 Possible improvements in the QHO-IPM

This study applies Tencent population mobility data to model and simulate intercity population mobility and verifies the feasibility and effectiveness of the QHO-IPM. In future research, the following limitations should be further considered:

For research data, most static statistical data, such as traditional population mobility census data and statistical yearbooks, only reflects the general state of population mobility. These data are updated slowly and with poor timeliness, with a temporal resolution of more than one year. In contrast, Tencent population mobility data record the cities where people move in (out) at a more fine-grained spatiotemporal resolution, reflecting the dynamic process of population mobility between different cities and providing data support for modeling and simulating intercity population mobility at a fine-grained spatiotemporal resolution. However, Tencent population mobility data have shortcomings in regard to data integrity. Despite the popularity of smartphones, they still do not fully cover all regions (especially remote mountainous areas) and populations (especially children, elderly individuals, etc.), which has a certain impact on the integrity of the data. The population mobility formed by some user groups who have not been connected to the Tencent platform has not been recorded, which inevitably leads to certain errors in the research results. In addition, due to the temporal resolution, some population mobility across two adjacent time units may be decomposed, making it impossible to obtain their origin and destination. This also influences the state and spatiotemporal pattern of intercity population mobility. Therefore, integrating vehicle GPS positioning data, railway travel data, and air travel data related to intercity population mobility and using these multimode intercity population mobility data for more accurate model training, parameter estimation, and result simulation are promising future research topics.

The correlation analysis between intercity population mobility and its driving factors should also be strengthened. QHO-IPM models intercity population mobility from the perspective of coupling driving factors, and it is finally solved by use of an optimized strategy. The driving factors have not yet been analyzed in depth. These driving factors are the key to promoting intercity population mobility, which may be of great significance in revealing the evolution pattern and impact mechanism of intercity population mobility. Therefore, we need to further strengthen the correlation analysis between intercity population mobility and its driving factors to verify the role of QHO-IPM in revealing the mechanism of intercity population mobility. Moreover, the construction of a dataset of the coupling relationship and coupling strength of the driving factors and the further exploration of the travel patterns and laws of intercity population mobility are future research objects. We will also strive to carry out more related research to further explore the potential of quantum models in the modeling and simulation of intercity population mobility.

7 Conclusions

This paper is primarily focused on the simulation of intercity population mobility as affected by complex coupling driving factors, and in the paper a quantum harmonic oscillator model for the simulation of intercity population mobility (QHO-IPM) is proposed. QHO-IPM realizes the correlation modeling between intercity population mobility and driving factors through a coupled quantum harmonic oscillator system. Optimization strategies are adopted to realize the optimal solution of the QHO-IPM. The case study shows that compared with the reference methods (ARIMA and LSTM), QHO-IPM achieves better simulation performance that results in small and stable simulation errors. Furthermore, the significant “one dominant and multiple auxiliary” factor pattern of the driving factors is revealed. In other words, QHO-IPM is not only of great importance for deepening the understanding of intercity population mobility and its laws but also for helping to provide a basis for intercity population mobility control from the perspective of driving factors.

Acknowledgments

We are grateful to the Tencent location big data platform for providing the data used in this study. Data source: The Tencent location big data platform (https://heat.qq.com/). We also wish to acknowledge WANG Jiaxian for collecting experimental data and for helpful comments on the paper.

Conflicts of interest

The authors declare no conflicts of interest.

Data availability statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

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In the past four decades, due to different research contents and spatial governance priorities, the names and scopes of regions such as Beijing-Tianjin-Hebei, Beijing-Tianjin-Tangshan, Bohai Rim and Greater Bohai Sea have changed many times. As the earliest humanities and economic geography research in China, its object area has attracted more and more attention such as disciplines of economic trade, ecological environment, and urban and rural planning. Based on the academic papers, monographs, and influential scientific research projects, this article reviews the research progress of the Beijing-Tianjin-Hebei region in the past 40 years. The progress has experienced a change process of "Beijing-Tianjin-Tangshan - Bohai Rim region - Beijing-Tianjin-Hebei". There are four recognizable phases in the research development to date. In the 1980s, economic geography mainly focused on land planning in the Beijing-Tianjin-Tangshan region, which was relatively limited in scale. In the 1990s, the research area was expanded to the Bohai Rim region, and the intersection of economic and trade science and geography was carried out in the process of economic integration in the eastern (northern) sub-regions. In the first decade of the 21st century, the research field turned to the integration of the Beijing-Tianjin-Hebei region, ecological environment science and urban planning science with large-scale intervention. In the 2010s, we started multidisciplinary regional comprehensive research on the coordinated development of the Beijing-Tianjin-Hebei region. During this period, the Chinese government carried out a series of major plans in the region, including the Beijing-Tianjin-Tangshan Land Planning in the 1980s, the Bohai Rim Economic Cooperation Zone in the 1990s, the Beijing-Tianjin-Hebei Metropolitan Region in the 2000s, and the Guidelines for the Coordinated Development of Beijing-Tianjin-Hebei Region in the 2010s. These major plans have formed a benign interactive relationship with regional research. This interactive relationship not only significantly enhances the scientific nature of regional planning and strategic decision-making, but also effectively promotes the development of humanities and economic geography, and it has also enhanced the research on the evolutionary laws of regional complex systems under the strong interaction between human and nature.

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We propose a possible scheme to study the thermalization in a quantum harmonic oscillator with random disorder. Our numerical simulation shows that through the effect of random disorder, the system can undergo a transition from an initial nonequilibrium state to a equilibrium state. Unlike the classical damped harmonic oscillator where total energy is dissipated, total energy of the disordered quantum harmonic oscillator is conserved. In particular, at equilibrium the initial mechanical energy is transformed to the thermodynamic energy in which kinetic and potential energies are evenly distributed. Shannon entropy in different bases are shown to yield consistent results during the thermalization.

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Industrial agglomeration is a highly prominent geographical feature of economic activities, and it is an important research topic in economic geography. However, mechanism-based explanations of industrial agglomeration often differ due to a failure to distinguish properly between the spatial distribution of industries and the stages of industrial agglomeration. Based on micro data from three national economic censuses, this study uses the Duranton-Overman (DO) index method to calculate the spatial distribution of manufacturing industries (three-digit classifications) in the Beijing-Tianjin-Hebei region (BTH region hereafter) from 2004 to 2013 as well as the hurdle model to explain quantitatively the influencing factors and differences in the two stages of agglomeration formation and agglomeration development. The research results show the following: (1) In 2004, 2008, and 2013, there were 124, 127, and 129 agglomerations of three-digit industry types in the BTH region, respectively. Technology-intensive and labor-intensive manufacturing industries had high agglomeration intensity, but overall agglomeration intensity declined during the study period, from 0.332 to 0.261. (2) There are two stages of manufacturing agglomeration, with different dominant factors. During the agglomeration formation stage, the main locational considerations of enterprises are basic conditions. Agricultural resources and transportation have negative effects on agglomeration formation, while labor pool and foreign investment have positive effects. In the agglomeration development stage, enterprises focus more on factors such as agglomeration economies and policies. Internal and external industry linkages both have a positive effect, with the former having a stronger effect, while development zone policies and electricity, gas, and water resources have a negative effect. (3) Influencing factors on industrial agglomeration have a scale effect, and they all show a weakening trend as distance increases, but different factors respond differently to distance.

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Quantum computers are an emerging technology promising to be vastly more powerful at solving certain problems than any conventional computer. These devices are now moving out of the laboratory and becoming generally programmable. This allows identical quantum tasks or algorithms to be implemented on radically different technologies to inform further development and scaling. We run a series of algorithms on the two leading platforms: trapped atomic ions and superconducting circuits. Whereas the superconducting system offers faster gate clock speeds and a solid-state platform, the ion-trap system features superior qubits and reconfigurable connections. The performance of these systems is seen to reflect the topology of connections in the base hardware, supporting the idea that quantum computer applications and hardware should be codesigned.

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Cities and villages are components of a specific organism. Only the sustainable development of two parts can support the prosperous development as a whole. According to the theory of man-earth areal system, urban-rural integrated system and rural regional system are the theoretical bases for entirely recognizing and understanding urban-rural relationship. To handle the increasingly severe problems of "rural disease" in rapid urbanization, accelerating rural revitalization in an all-round way is not only a major strategic plan for promoting the urban-rural integration and rural sustainable development, but also a necessary requirement for solving the issues related to agriculture, rural areas, and rural people in the new era and securing a decisive victory in building a moderately prosperous society in all respects. This study explores the basic theories of urban-rural integration and rural revitalization and analyzes the main problems and causes of rural development in the new era, proposing problem-oriented scientific approaches and frontier research fields of urban-rural integration and rural revitalization in China. Results show that the objects of urban-rural integration and rural revitalization is a regional multi-body system, which mainly includes urban-rural integration, rural complex, village-town organism, and housing-industry symbiosis. Rural revitalization focuses on promoting the reconstruction of urban-rural integration system and constructs a multi-level goal system including urban-rural infrastructure networks, zones of rural development, fields of village-town space and poles of rural revitalization. Currently, the rural development is facing the five problems: high-speed non-agricultural transformation of agriculture production factors, over-fast aging and weakening of rural subjects, increasingly hollowing and abandoning of rural construction land, severe fouling of rural soil and water environment and deep pauperization of rural poverty-stricken areas. The countryside is an important basis for the socioeconomic development in China, and the strategies of urban-rural integration and rural revitalization are complementary. The rural revitalization focuses on establishing the institutional mechanism for integrated urban-rural development and constructs the comprehensive development system of rural regional system, which includes transformation, reconstruction and innovation in accordance with the requirements of thriving businesses, pleasant living environments, social etiquette and civility, effective governance, and prosperity. Geographical research on rural revitalization should focus on the complexity and dynamics of rural regional system and explore new schemes, models and scientific approaches for the construction of villages and towns, which are guided by radical cure of "rural disease", implement the strategy of rural revitalization polarization, construct the evaluation index system and planning system of rural revitalization, thus providing advanced theoretical references for realizing the revitalization of China's rural areas in the new era.

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Mazzoli

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et al., 2019. Field theory for recurrent mobility. Nature Communications, 10(1): 3895.

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Cited in this article [1] Abstract

Understanding human mobility is crucial for applications such as forecasting epidemic spreading, planning transport infrastructure and urbanism in general. While, traditionally, mobility information has been collected via surveys, the pervasive adoption of mobile technologies has brought a wealth of (real time) data. The easy access to this information opens the door to study theoretical questions so far unexplored. In this work, we show for a series of worldwide cities that commuting daily flows can be mapped into a well behaved vector field, fulfilling the divergence theorem and which is, besides, irrotational. This property allows us to define a potential for the field that can become a major instrument to determine separate mobility basins and discern contiguous urban areas. We also show that empirical fluxes and potentials can be well reproduced and analytically characterized using the so-called gravity model, while other models based on intervening opportunities have serious difficulties.

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A key open question in quantum computing is whether quantum algorithms can potentially offer a significant advantage over classical algorithms for tasks of practical interest. Understanding the limits of classical computing in simulating quantum systems is an important component of addressing this question. We introduce a method to simulate layered quantum circuits consisting of parametrized gates, an architecture behind many variational quantum algorithms suitable for near-term quantum computers. A neural-network parametrization of the many-qubit wavefunction is used, focusing on states relevant for the Quantum Approximate Optimization Algorithm (QAOA). For the largest circuits simulated, we reach 54 qubits at 4 QAOA layers, approximately implementing 324 RZZ gates and 216 RX gates without requiring large-scale computational resources. For larger systems, our approach can be used to provide accurate QAOA simulations at previously unexplored parameter values and to benchmark the next generation of experiments in the Noisy Intermediate-Scale Quantum (NISQ) era.

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Shi

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Using Tencent location big data, this study analyzes the spatial pattern of population flow among 368 cities in China and identifies the influencing factors related to population inflow and outflow based on an exponential random graph model (ERGM). 1) From 2015 to 2018, the spatial distribution pattern of population flow was relatively stable, forming a rhombic spatial structure with Beijing, Shenzhen, Shanghai, Guangzhou, Chengdu, and Dongguan as the ‘center’. The densely populated nodes and channels are mainly concentrated to the east of the Hu Huanyong Line. The significance of this study lies in further determining the core cities and main pillars in the population flow network. 2) The urban subgroup structure obtained by community division shows obvious geographical proximity and inter-provincial differentiation among communities, which form not only small urban subgroups with the provincial capital city forming the core and bordered by the provincial boundary, but also large urban subgroups with a multi-center structure spanning provincial administrative boundaries. However, for most cities, the provincial boundary delimits the main flow circle, and population flow within the same province is more frequent. 3) The influencing factors of the population inflow and outflow networks determined by the ERGM model are consistent with the predictions of neoclassical economics. Market and economic factors such as population scale, urbanization level, time cost, and economic cost still play a leading role in population flow. 4) The attraction of a city to the floating population depends on its individual attributes, while the urban population outflow depends more on the external-network-related elements of the city. To a certain extent, this study verifies the predominance of the urban “pull” in the push-pull theory and the comprehensive effect of various distance factors.

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Cited in this article [2] Abstract

The movements of individuals within and among cities influence critical aspects of our society, such as well-being, the spreading of epidemics, and the quality of the environment. When information about mobility flows is not available for a particular region of interest, we must rely on mathematical models to generate them. In this work, we propose Deep Gravity, an effective model to generate flow probabilities that exploits many features (e.g., land use, road network, transport, food, health facilities) extracted from voluntary geographic data, and uses deep neural networks to discover non-linear relationships between those features and mobility flows. Our experiments, conducted on mobility flows in England, Italy, and New York State, show that Deep Gravity achieves a significant increase in performance, especially in densely populated regions of interest, with respect to the classic gravity model and models that do not use deep neural networks or geographic data. Deep Gravity has good generalization capability, generating realistic flows also for geographic areas for which there is no data availability for training. Finally, we show how flows generated by Deep Gravity may be explained in terms of the geographic features and highlight crucial differences among the three considered countries interpreting the model's prediction with explainable AI techniques.© 2021. The Author(s).

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2020 was an unprecedented year, with rapid and drastic changes in human mobility due to the COVID-19 pandemic. To understand the variation in commuting patterns among the Chinese population across stable and unstable periods, we used nationwide mobility data from 318 million mobile phone users in China to examine the extreme fluctuations of population movements in 2020, ranging from the Lunar New Year travel season (chunyun), to the exceptional calm of COVID-19 lockdown, and then to the recovery period. We observed that cross-city movements, which increased substantially in chunyun and then dropped sharply during the lockdown, are primarily dependent on travel distance and the socio-economic development of cities. Following the Lunar New Year holiday, national mobility remained low until mid-February, and COVID-19 interventions delayed more than 72.89 million people returning to large cities. Mobility network analysis revealed clusters of highly connected cities, conforming to the social-economic division of urban agglomerations in China. While the mass migration back to large cities was delayed, smaller cities connected more densely to form new clusters. During the recovery period after travel restrictions were lifted, the netflows of over 55% city pairs reversed in direction compared to before the lockdown. These findings offer the most comprehensive picture of Chinese mobility at fine resolution across various scenarios in China and are of critical importance for decision making regarding future public-health-emergency response, transportation planning and regional economic development, among others.

[50]

Tang

, Zhang

, Wang

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https://doi.org/10.11821/dlyj020190422

Cited in this article [1] Abstract

Population movement is the main carrier of inter-city factor flow and resource alloca-tion. It is also one of the main forms of regional network construction. As the Yangtze River Delta integration has become a national strategy, in order to promote inter-city population movement and regional integration, it is of great importance to recognize the pattern of population movement and analyze the influencing factors. This study focuses on the inter-city daily mobility within 48 hours, one of the important components of population movement, based on the Weibo sign-in data. It analyzes the pattern of inter-city population movements in the Yangtze River Delta, applying the gravity model to test influencing factors, from perspectives of movement cost and city characteristics. The results indicate that: (1) population movements in the study area have multiple cores, and connections between cores and their hinterlands are relatively balanced. In addition, there are three communities in this region, and Shanghai, Suzhou and Hangzhou belong to the same community. (2) Strong population movements occur within each province, and population movement systems of the three provinces are different. Zhejiang and Jiangsu provinces have formed a relatively mature multi-level population mobility system, while the population mobility system in Anhui Province needs to be improved. (3) Movement cost and city characteristics complement the inter-city mobility model shaped by physical distances, and compared with movement cost, city characteristics have a greater impact on population movement. (4) Compared with the inter-province movement, population movements within each province are stronger, and are more likely to occur between cities with different cultures. The greater the differences between the two cities, regarding economic scale and administrative level, the stronger the population movements between them. Besides, differences in industrial structure will inhibit intercity mobility, while differences in education level can promote mobility. This paper expands the applications of the gravity model, analyzes inter-city daily movement mechanism, and provides references for understanding the process of the Yangtze River Delta integration, as well as optimizes policies from the perspective of daily population movement.

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et al., 2019. An extended exploration and preferential return model for human mobility simulation at individual and collective levels. Physica A: Statistical Mechanics and Its Applications, 534: 121921.

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The effective modeling of urban growth is crucial for urban planning and analyzing the causes of land-use dynamics. As urbanization has slowed down in most megacities, improved urban growth modeling with minor changes has become a crucial open issue for these cities. Most existing models are based on stationary factors and spatial proximity, which are unlikely to depict spatial connectivity between regions. This research attempts to leverage the power of real-world human mobility and consider intra-city spatial interaction as an imperative driver in the context of urban growth simulation. Specifically, the gravity model, which considers both the scale and distance effects of geographical locations within cities, is employed to characterize the connection between land areas using individual trajectory data from a macro perspective. It then becomes possible to integrate human mobility factors into a neural-network-based cellular automata (ANN-CA) for urban growth modeling in Beijing from 2013 to 2016. The results indicate that the proposed model outperforms traditional models in terms of the overall accuracy with a 0.60% improvement in Cohen’s Kappa coefficient and a 0.41% improvement in the figure of merit. In addition, the improvements are even more significant in districts with strong relationships with the central area of Beijing. For example, we find that the Kappa coefficients in three districts (Chaoyang, Daxing, and Shunyi) are considerably higher by more than 2.00%, suggesting the possible existence of a positive link between intense human interaction and urban growth. This paper provides valuable insights into how fine-grained human mobility data can be integrated into urban growth simulation, helping us to better understand the human-land relationship.

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Wang

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Through the construction of a population flow and migration relationship matrix, this paper analyzes population flow and migration in the Yangtze River Delta urban agglomeration during the Spring Festival travel rush and daily period. This paper also studies the urban network spatial structure characteristics and the influencing factors from the perspective of inter-provincial population flow and migration. The results show the following: (1) as a central city, Shanghai has a significant siphon effect, with Suzhou, Nanjing, Hangzhou, Ningbo, Wuxi and Changzhou accumulating 86.95% of the incoming population. The Shanghai–Jiangsu cross-border floating population is active and accounts for 40.83% of the total mobility scale in the same period. The population flow and migration network in the Yangtze River Delta urban agglomeration shows obvious hierarchical characteristics. The secondary network relationship during the Spring Festival travel rush is the main migration path, while the first-level network relationship in the daily period is the main flow path. (2) Three indicators, namely, the network density, mean centrality, and control force based on the population flow and migration, consistently show that the Yangtze River Delta urban agglomeration network presents a strong connection state with the formation of a local cluster structure, highlighting that the city tightness in terms of population flow and migration also has dual attributes, which refers to “the restriction of the geographic space effect” and “overcoming the friction of space”. (3) Economic scale, political resources, industrial structure, and the historical basis are important factors influencing the formation of population flows and migration networks. Employment opportunities and labor wages are key guiding factors of the population migration direction, and spatial distance is a conditional factor influencing the formation of population flows and migration networks. The inter-provincial boundary, temporal distance, and transboundary frequency are the decisive factors for the formation of network patterns of population flow and migration.

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Cited in this article [2] Abstract

Studies of human mobility in the past decade revealed a number of general scaling laws. However, to reproduce the scaling behaviors quantitatively at both the individual and population levels simultaneously remains to be an outstanding problem. Moreover, recent evidence suggests that spatial scales have a significant effect on human mobility, raising the need for formulating a universal model suited for human mobility at different levels and spatial scales. Here we develop a general model by combining memory effect and population-induced competition to enable accurate prediction of human mobility based on population distribution only. A variety of individual and collective mobility patterns such as scaling behaviors and trajectory motifs are accurately predicted for different countries and cities of diverse spatial scales. Our model establishes a universal underlying mechanism capable of explaining a variety of human mobility behaviors, and has significant applications for understanding many dynamical processes associated with human mobility.

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Yan

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https://royalsocietypublishing.org/doi/10.1098/rsif.2014.0834

Cited in this article [2] Abstract

Despite the long history of modelling human mobility, we continue to lack a highly accurate approach with low data requirements for predicting mobility patterns in cities. Here, we present a population-weighted opportunities model without any adjustable parameters to capture the underlying driving force accounting for human mobility patterns at the city scale. We use various mobility data collected from a number of cities with different characteristics to demonstrate the predictive power of our model. We find that insofar as the spatial distribution of population is available, our model offers universal prediction of mobility patterns in good agreement with real observations, including distance distribution, destination travel constraints and flux. By contrast, the models that succeed in modelling mobility patterns in countries are not applicable in cities, which suggests that there is a diversity of human mobility at different spatial scales. Our model has potential applications in many fields relevant to mobility behaviour in cities, without relying on previous mobility measurements.

[62]

Yang

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, 2022. The innovative geographical foundation of the relative decline of economic growth in Beijing-Tianjin-Hebei region. Acta Geographica Sinica, 77(6): 1320-1338. (in Chinese)

https://doi.org/10.11821/dlxb202206003

Cited in this article [1] Abstract

How to raise innovation efficiency to promote the economic development from relative decline to prosperity and then drive the development of the northern hinterland becomes a prominent problem facing the in-depth acceleration of the coordinated development of the Beijing-Tianjin-Hebei (BTH) region in the new development stage. This paper points out that the inefficiency of innovation is the key to economic stagnation in the BTH region, and empirically demonstrates that the relative inefficiency of innovation geography in the region is the main reason that restricts the innovative efficiency development based on the "Density, Distance, Division, Differentiation" (4D) framework. The strategic proposition of reshaping innovation geography in the BTH region is put forward. First of all, the article analyzes the disparities in the evolution trends of economic efficiency, innovation competitiveness and innovation efficiency in the BTH region, the Yangtze River Delta, the Guangdong-Hong Kong-Macao region, and finds that the economic efficiency in the BTH region is lack of competitiveness and tends to decline, characterized by weak innovative advantage and low innovation efficiency. Secondly, based on the 4D framework, there are several reasons for the relative inefficiency of innovation from a perspective of economic geography. These reasons include the coexistence of low density of economy, population and patent with high concentration imbalance, the wide scope of patent transfer and the low localization level of industry-university-research collaboration, and the severe innovation segmentation with strong spatial heterogeneity, and relatively weak heterogeneous advantage. Finally, the measurement results show that the relative change of 4D factors will significantly affect the fluctuation of innovation relative efficiency, which indicates that the relative inefficiency of innovation geography in the BTH region is the root of innovation relative inefficiency. This paper shows that the reconstruction of innovation economic geography based on 4D framework is vital to improve innovation efficiency and realize innovation-driven development in the BTH region.

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https://linkinghub.elsevier.com/retrieve/pii/S0191261521000369

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[66]	Yue T, Liu Y, Du Z et al., 2022. Quantum machine learning of eco-environmental surfaces. Science Bulletin, 67(10): 1031-1033. https://doi.org/10.1016/j.scib.2022.02.009 https://www.ncbi.nlm.nih.gov/pubmed/36546246 Cited in this article [1]

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[68]	Zelinsky W, 1971. The hypothesis of the mobility transitio. Geographical Review, 61: 219-249. https://doi.org/10.2307/213996 https://www.jstor.org/stable/213996?origin=crossref Cited in this article [1]

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[70]	Zhang J, Zheng Y, Qi D et al., 2018. Predicting citywide crowd flows using deep spatio-temporal residual networks. Artificial Intelligence, 259: 147-166. https://doi.org/10.1016/j.artint.2018.03.002 https://linkinghub.elsevier.com/retrieve/pii/S0004370218300973 Cited in this article [1]

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https://www.ncbi.nlm.nih.gov/pubmed/36307471

Cited in this article [1] Abstract

Globally, since the outbreak of the Omicron variant in November 2021, the number of confirmed cases of COVID-19 has continued to increase, posing a tremendous challenge to the prevention and control of this infectious disease in many countries. The global daily confirmed cases of COVID-19 between November 1, 2021, and February 17, 2022, were used as a database for modeling, and the ARIMA, MLR, and Prophet models were developed and compared. The prediction performance was evaluated using mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE). The study showed that ARIMA (7, 1, 0) was the optimum model, and the MAE, MAPE, and RMSE values were lower than those of the MLR and Prophet models in terms of fitting performance and forecasting performance. The ARIMA model had superior prediction performance compared to the MLR and Prophet models. In real-world research, an appropriate prediction model should be selected based on the characteristics of the data and the sample size, which is essential for obtaining more accurate predictions of infectious disease incidence.© 2022. The Author(s).

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Cited in this article [2] Abstract

Simulations based on spatial interaction models have been widely applied to understand the strength of relationships between geographical elements, but many issues remain unclear and deviations between actual and simulated results have often been seriously underestimated. A high-precision Baidu migration process combined with mass relationships is applied in this study and enables the generation of regression coefficients of gravity model based on programmed large-scale regression simulations. A series of accuracy assessments are then developed for 2015 empirical projection daily regression coefficients that can be applied to Chinese spring interprovincial mobile gravity model variables as well as spatiotemporal research that utilizes regression coefficients within a heterogeneity research model. This approach also enables the error within the gravity model to be assessed in terms of floating population simulations. The results of this analysis lead to a number of clear conclusions, including the fact that parameter calibration complexity for the Chinese population mobility gravity model is reflected in the degree of influence asymmetry within spatial object interaction variables, and that the spatial heterogeneity of the variable regression coefficient increases in two distinct fashions. The first of these increases has to do with the overall influence of specific variables, including the fact that differences between proxies tend to be higher than inflow-outflow characteristics. In contrast, the second set of increases is related to economic levels, industrial scales, the proportion of the tertiary industry, and public service facilities. In this latter case, two-way population flow exerts a more profound influence on results and thus the scope of possible explanations for phenomena is more extensive. The regression coefficient for the existence of positive and negative proxy variables therefore relates to differences in spatial heterogeneity, including at the city level, and also assumes that floating population gravity model regression coefficients ignore spatiotemporal changes in the heterogeneity coefficient. This leads to spatial differences in estimated results and thus convergence trends, but further enables the identification of anisotropic interactions in extension space. The second main conclusion of this research is that the national scale population flow distance attenuation coefficient was 1.970 during the spring of 2015, while at the level of prefectural administrative units and given population outflow, the range encapsulated by this coefficient fell between 0.712 (Zhumadian) and 7.699 (Urumqi). Data also reveal a population inflow coefficient of 0.792 for this year that ranged as high as 8.223 in both Sanya and Urumqi. Population flow simulation results using the gravity model and including Baidu migration measured flow data were also subject to significant error. Third, the results of this analysis reveal a total fitting error of 85.54% in weighted absolute mean; the spatial interaction effect within this is responsible for a maximum error of 86.09% in actual and simulated flows, while relative outflow force and attractiveness encompass 57.73% and 49.34% of model error, respectively. These results show that the spatial interaction effect remains most difficult to model in terms of current factors.

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[75]	Zúñiga J, Bastida A, Requena A, 2017. Quantum solution of coupled harmonic oscillator systems beyond normal coordinates. Journal of Mathematical Chemistry, 55(10): 1964-1984. https://doi.org/10.1007/s10910-017-0777-1 http://link.springer.com/10.1007/s10910-017-0777-1 Cited in this article [1]

Funding

National Natural Science Foundation of China(42230406)

National Natural Science Foundation of China(42130103)

National Natural Science Foundation of China(41971404)

National Natural Science Foundation of China(42201504)

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Abstract
Key words
Cite this article
1 Introduction
2 Related works
2.1 Driving factors of population mobility
2.2 The intercity population mobility simulation model
2.3 Quantum computing and its applications
3 Basic concept
Figure 1 The overall framework of QHO-IPM
4 Methodology
4.1 The coupled quantum oscillator system
4.2 Decoupling the coupled quantum oscillator system
4.3 Optimization and solution of the quantum harmonic oscillator model
4.4 Mapping time-varying probability to population mobility
5 Implementation and results
5.1 Data description and experimental configuration
Figure 2 Study area and average intercity population mobility
Table 1 Definition of evaluation criteria
5.2 Simulation results of intercity population mobility
Figure 3 The simulated intercity population mobility using different methods
Figure 4 The 1:1 diagram between the simulated intercity population mobility and the observed intercity population mobility
Table 2 The evaluation indicators of different methods
5.3 Error analysis
Figure 5 Error distribution of the different methods
6 Discussion
6.1 Influence degree of the different factors
Figure 6 Distribution of the monthly influence degree of different factors
6.2 Possible improvements in the QHO-IPM
7 Conclusions
Acknowledgments
Conflicts of interest
Data availability statement
References
Funding

Received	Accepted	Published
2022-12-10	2023-10-23	2024-03-25
Issue Date
2024-04-24

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Abstract

Key words

Cite this article

1 Introduction

2 Related works

2.1 Driving factors of population mobility

2.2 The intercity population mobility simulation model

2.3 Quantum computing and its applications

3 Basic concept

Figure 1 The overall framework of QHO-IPM

4 Methodology

4.1 The coupled quantum oscillator system

4.2 Decoupling the coupled quantum oscillator system

4.3 Optimization and solution of the quantum harmonic oscillator model

4.4 Mapping time-varying probability to population mobility

5 Implementation and results

5.1 Data description and experimental configuration

Figure 2 Study area and average intercity population mobility

Table 1 Definition of evaluation criteria

5.2 Simulation results of intercity population mobility

Figure 3 The simulated intercity population mobility using different methods

Figure 4 The 1:1 diagram between the simulated intercity population mobility and the observed intercity population mobility

Table 2 The evaluation indicators of different methods

5.3 Error analysis

Figure 5 Error distribution of the different methods

6 Discussion

6.1 Influence degree of the different factors

Figure 6 Distribution of the monthly influence degree of different factors

6.2 Possible improvements in the QHO-IPM

7 Conclusions

Acknowledgments

Conflicts of interest

Data availability statement

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References

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Footnotes

Funding