Lattice boltzmann flow field simulation method based on orthogonal moment space quantum bgk collision
Patent Information
- Application Number
- CN202611214787.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Priority Date
- 2026-04-30
- Filing Date
- 2026-08-11
- Publication Date
- 2026-10-27
AI Technical Summary
[0007]本发明的目的在于提供一种基于正交矩空间量子BGK碰撞的格子玻尔兹曼流场模拟方法，以解决在利用量子计算模块进行黏性流体模拟时，BGK碰撞的非幺正耗散过程难以由量子门直接实现，以及量子碰撞结果未能与流体黏性参数、分布函数迁移、边界条件处理和宏观流场更新形成完整联系的问题
1. 通过正交矩空间表达将BGK碰撞算子的谱结构显式化，能够清楚区分BGK过程中守恒模态与非守恒模态的作用机制；
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Abstract
Description
Technical Field
[0001] This invention relates to the interdisciplinary fields of quantum computing and computational fluid dynamics, specifically to a lattice Boltzmann flow field simulation method based on orthogonal moment space quantum BGK collisions. The method constructs lattice Boltzmann distribution functions and equilibrium distribution functions based on the density, velocity, and viscosity parameters of the fluid to be simulated. It then utilizes a quantum computing module to perform BGK collision updates in orthogonal moment space while maintaining mass and momentum conservation. Finally, by combining distribution function migration, boundary condition processing, and macroscopic physical quantity reconstruction, the evolution of the fluid density and velocity fields over time is obtained. Background Technology
[0002] Computational fluid dynamics (CFD) is an engineering analysis method that solves fluid control equations through numerical discretization and computer simulation. It is widely used in aerodynamic analysis, flow and heat transfer calculations, and prediction of complex flow fields. Traditional methods typically solve the Navier-Stokes equations directly by discretization, while the lattice Boltzmann method, from the perspective of mesoscopic statistical physics, recovers the macroscopic behavior of fluids through the collision and migration process of the distribution function on the discrete velocity set. It has advantages such as regular algorithm structure, strong parallelism, and ease of handling complex boundaries, and has received widespread attention in fluid numerical simulation in recent years.
[0003] In the lattice Boltzmann method, the single-relaxation-time BGK collision model is one of the most commonly used collision models due to its concise expression and ease of implementation. The core idea of this model is to relax the discrete distribution function towards the equilibrium distribution function with a given relaxation rate, thereby achieving the dissipative evolution of non-equilibrium information towards an equilibrium state. For each time step, the BGK collision process requires updating the distribution function in each discrete velocity direction. As the dimension of the discrete velocity and the size of the spatial grid increase, this process generates a large number of repetitive local numerical operations; therefore, its efficient implementation has always been one of the key issues in lattice Boltzmann method research.
[0004] Meanwhile, quantum computing, with its properties such as quantum superposition and quantum parallelism, exhibits potential advantages in specific types of linear algebraic operations and state evolution problems, thus providing a new research direction for the reconstruction of numerical computation methods for fluid mechanics. However, existing research on quantum fluid computing largely focuses on solving quantum linear equations, quantum variational algorithms, or the overall replacement of classical numerical processes, while research on the quantum expression of fundamental operators within the lattice Boltzmann method, especially the BGK collision process itself, remains relatively limited. Furthermore, the BGK collision process is essentially a class of non-unitary contraction mappings with dissipative characteristics. The classical BGK model requires that conserved modes remain unchanged while non-conserved modes decay according to a given relaxation rate. Since quantum gate evolution must satisfy global unitarity, this non-unitary relaxation process is difficult to directly realize in quantum circuits. Therefore, how to transform this collision process into an evolutionary form that can be realized by quantum systems while maintaining the physical mechanism of BGK becomes the core problem that needs to be solved when introducing the lattice Boltzmann collision step into the quantum computing framework.
[0005] In existing technologies, the processing of BGK collisions typically remains at the level of classical velocity space updates, lacking a method that can explicitly distinguish between conserved and non-conserved modes and express collision relaxation in a structured manner within a quantum system. Furthermore, fluid BGK collisions exhibit a distinct conservation-dissipation structure. Conserved quantities corresponding to fluid mass and momentum should remain unchanged during local collisions, while non-conserved quantities corresponding to viscous stress and higher-order non-equilibrium information should decay according to the relaxation rate determined by the viscous parameters. Directly constructing a unified quantum transformation within the original discrete velocity space makes it difficult to clearly distinguish between these two types of physical quantities and easily degenerates the quantum collision process into matrix operations lacking specific fluid physics constraints.
[0006] Flow field simulations also require the post-collision distribution function to be used for spatial migration, boundary condition processing, and macroscopic physical quantity reconstruction in the next time step. Simply obtaining the mathematical mapping results of the local collision operator is insufficient to independently represent the time-propagation process of the lattice Boltzmann flow field. Therefore, a method combining fluid density, velocity, viscosity parameters, and boundary conditions is needed to realize conservation and dissipation in BGK collisions under quantum gate unitary evolution constraints, and to continue using the quantum collision results for distribution function migration and macroscopic flow field updates. Summary of the Invention
[0007] The purpose of this invention is to provide a lattice Boltzmann flow field simulation method based on orthogonal moment space quantum BGK collisions, in order to solve the problems that when using quantum computing modules to simulate viscous fluids, the non-unitary dissipation process of BGK collisions is difficult to be directly realized by quantum gates, and the quantum collision results fail to form a complete connection with fluid viscosity parameters, distribution function migration, boundary condition processing and macroscopic flow field updates.
[0008] This invention maps the non-equilibrium deviation of the distribution function relative to the local equilibrium distribution function to an orthogonal moment space, thus separating the conserved modes corresponding to fluid mass and momentum from the non-conserved modes corresponding to viscous stress and higher-order non-equilibrium information. While keeping the conserved modes unchanged, a controlled rotation involving auxiliary qubits is applied to the non-conserved modes based on the BGK relaxation rate of the fluid to be simulated. The non-equilibrium distribution after the collision is obtained from the ground state branch of the auxiliary qubits. Subsequently, the distribution function is reconstructed, migrated, boundary processed, and macroscopic quantities are updated to obtain the fluid density and velocity at the next time step.
[0009] The present invention adopts the following technical solution:
[0010] A quantum implementation method for an orthogonal moment space BGK collision operator, comprising the following steps: Step 1, Flow field state setting: Discretize the fluid region to be simulated by grid points, select the lattice Boltzmann discrete velocity model, obtain or set the initial density, initial velocity, viscosity parameters and boundary conditions of the fluid to be simulated, and initialize the distribution function of each grid point in different discrete velocity directions; Step 2, Calculation of macroscopic quantities and equilibrium state: Determine the fluid density based on the zero moment of the distribution function at each grid point in the current time step, and determine the fluid momentum and fluid velocity based on the first moment of the distribution function; determine the local equilibrium state distribution function based on the fluid density and fluid velocity, and determine the BGK collision relaxation rate based on the viscosity parameters of the fluid to be simulated; Step 3, Construction and quantum encoding of non-equilibrium deviation: Calculate the difference between the distribution function and the local equilibrium state distribution function to obtain the non-equilibrium deviation characterizing the non-equilibrium state of the fluid; encode the non-equilibrium deviation into the system register of the quantum computing module, and initialize the auxiliary qubit to the ground state; Step 4, Orthogonal Moment Space Transformation: Perform orthogonal moment transformation on the system register to map the non-equilibrium deviation from the discrete velocity space to the orthogonal moment space; divide the modes in the orthogonal moment space into conserved modes corresponding to fluid mass and momentum, and non-conserved modes corresponding to viscous stress and higher-order non-equilibrium information; Step 5, Quantum BGK Collision: Keeping the conserved mode unchanged, determine the rotation parameters of the controlled rotation operation of the auxiliary qubit according to the BGK collision relaxation rate; apply controlled rotation to the auxiliary qubit under the control of the non-conserved mode so that when the auxiliary qubit is in the ground state branch, the preserved amplitude of the non-conserved mode corresponds to the relaxation factor of the BGK collision. Step 6: Collision Result Extraction and Distribution Function Reconstruction: Measure the auxiliary qubit and retain the system register state corresponding to the ground state of the auxiliary qubit measurement result; perform an inverse orthogonal moment transformation on the system register state to obtain the non-equilibrium deviation after the collision; add the non-equilibrium deviation after the collision to the local equilibrium state distribution function to obtain the post-collision distribution function of each lattice point; Step 7, Migration and Boundary Processing: Based on the discrete velocity direction corresponding to each distribution function, the post-collision distribution function is migrated to adjacent grid points, and the distribution function at the boundary position of the computational domain is processed according to the preset boundary conditions to obtain the distribution function of each grid point in the next time step; Step 8: Macroscopic flow field update: Based on the distribution function of each grid point in the next time step, redetermine the fluid density and fluid velocity, repeat steps S2 to S8 until the preset simulation time is reached, and output the density field and velocity field of the fluid region to be simulated.
[0011] The orthogonal transformation is implemented using an orthogonal matrix Q obtained by normalizing orthogonal polynomials on the discrete velocity set, satisfying that the transpose of Q is equal to the inverse of Q, so that the BGK collision operator is represented in the orthogonal moment space as a block diagonal structure with a conserved modal eigenvalue of 1 and a non-conserved modal eigenvalue that establishes a linear contraction relationship with the relaxation rate.
[0012] The BGK collision does not act directly on the distribution function itself, but on the deviation. By subtracting the equilibrium distribution function from the distribution function, the classical BGK collision process is rewritten as a linear contraction process of the deviation, which facilitates spectral decomposition and quantum realization.
[0013] The extended unitary evolution of the auxiliary qubit constructs a two-dimensional coupled subspace for each non-conservative mode, and applies a controlled rotation gate in the composite space composed of the system mode and the auxiliary qubit, so that the rotation angle parameter and the relaxation rate are correlated, so as to obtain an effective mapping consistent with the shrinkage coefficient of the non-conservative mode when the auxiliary qubit is kept in the ground state branch. The controlled rotation gate acts on the system mode register and the auxiliary qubit register.
[0014] The effective mapping is obtained by performing post-selection or equivalent offset processing on the joint evolution results of the system quantum state and the auxiliary quantum bit, and taking the component of the auxiliary quantum bit in the ground state as the effective collision result corresponding to the system state, thereby realizing the unification between the unitary evolution in the total space and the non-unitary BGK relaxation in the subsystem.
[0015] The method establishes a quantum realization process for the BGK collision step in the lattice Boltzmann method, excluding the quantum encoding of the migration step and the boundary condition step; wherein the calculation of the equilibrium distribution function and the reconstruction of the distribution function after the collision can be completed on classical computing devices.
[0016] Compared with the prior art, the present invention has the following advantages: 1. By making the spectral structure of the BGK collision operator explicit through orthogonal moment space representation, the interaction mechanisms of conserved and non-conserved modes in the BGK process can be clearly distinguished. 2. By using unitary expansion of auxiliary qubits, the non-unitary contraction process in the subsystem is transformed into a unitary evolution process in the overall system, thus solving the key difficulty that BGK collisions cannot be directly realized by quantum gates; 3. This method establishes a unified implementation framework of "orthogonal moment basis transformation - modal selective evolution - inverse transformation" based on a multi-time relaxation model, which has good physical interpretability and potential for further expansion; Attached Figure Description
[0017] Figure 1 This is a schematic diagram of the overall process of the orthogonal moment space BGK collision quantum realization method described in this invention.
[0018] Figure 2 This is a schematic diagram illustrating the principle of separating conserved and non-conserved modes in orthogonal moment space and the participation of auxiliary qubits in unitary expansion in this invention.
[0019] Figure 3 This is a schematic diagram of the controlled rotation quantum circuit for the auxiliary qubit corresponding to the non-conservative mode in this invention.
[0020] Figure 4 This is a comparison of simulation results for a one-dimensional isothermal density discontinuous flow field.
[0021] Figure 5 This is a simulation of a two-dimensional periodic domain square column scattering flow field. Detailed Implementation
[0022] The present invention will be further described below with reference to the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and should not be used to limit the scope of protection of the present invention.
[0023] This embodiment uses the D2Q9 discrete velocity model as an example to illustrate the implementation process of the present invention. As shown in Figure 1, for the Boltzmann equation considering the BGK collision model: ; First, define the distribution function vector in the discrete velocity space. and the corresponding equilibrium distribution function vector And define the deviation amount For the classic BGK collision model, collision updates can be written as adjustments to the deviation. The linear contraction process: ; ; in Indicates the relaxation rate. express The distribution function of direction, express The equilibrium distribution function of the direction. This indicates the deviation (unbalance).
[0024] Because this expression directly reveals that the collision is essentially a decay of the part deviating from the equilibrium state, it is more suitable for spectral analysis and quantum gate implementation than directly quantizing the distribution function itself. Joint environment bit / auxiliary bit encoding to Quantum states, consider the general superposition form: ; in, The normalization coefficient of the distribution function deviation. These are the basis vectors of the discrete velocity space. This represents the ground state of the auxiliary qubit. In the subscript, D indicates the system direction register, and E indicates the auxiliary bit register.
[0025] Using d'Humieres' linear relaxation model, an invertible unitary matrix of basis transformation is constructed using orthogonal polynomials on discrete velocities. After acting on the initial state, we have: ; The modes are divided into conserved modes. and non-conservative modes .
[0026] As shown in Figure 1, further, an orthogonal matrix is constructed in the discrete velocity space. And map the deviation to the orthogonal moment space: ; because Since it is an orthogonal matrix, its inverse transformation can be obtained by... The transpose of is given. In this orthogonal moment space, the BGK collision operator is represented in diagonal or block diagonal form, where conserved modes such as mass and momentum modes correspond to eigenvalue 1, which remains unchanged during the collision; the remaining non-conserved modes correspond to eigenvalues... and according to the relaxation rate Linear contraction occurs: .
[0027] Since the contraction mapping is not a unitary operator in the system space, such as Figure 2 As shown in Figure 3, this invention introduces an auxiliary qubit for each non-conservative mode and applies controlled rotations to the two-dimensional composite subspace composed of the system modes and the auxiliary qubits. In the D2Q9 model, there are typically 3 conserved modes (such as density and momentum modes) and 6 non-conservative modes. Therefore, Dir 0-2 in the quantum circuit maintains identical evolution, while Dir 3-8 correspond to the non-conservative modes. Each of these acts as a control bit to trigger a rotating gate that operates on the auxiliary bit Anc. By setting the rotation angle parameter θ, cosθ is made to correspond to the contraction coefficient of the non-conservative mode, i.e. After unitary evolution in total space, a portion of the system amplitude is transferred to the excitation branch of the auxiliary qubit. Meanwhile, it remains in the ground state branch of the auxiliary qubit. The components on the top precisely correspond to the modal contraction results required for classical BGK collisions. Here, the revolving door parameters... and Relationship satisfaction (based on (Definition of a door).
[0028] From a quantum mapping perspective, this invention does not require the direct construction of non-unitary gates in the system space, but rather achieves an equivalent collision process through the joint unitary evolution of the "system + auxiliary qubits". (Defining the operator) : ; Operator Momentary bases acting on the construction: ;
[0029] Rotation angle here With relaxation rate Satisfying Relationship: .
[0030] As shown in Figure 2, for the joint evolution results, a post-selection approach can be used, that is, retaining only the ground state branches of the auxiliary qubits to obtain a deterministic effective mapping. Alternatively, the process can be understood from the perspective of equivalent bias. This achieves a unification between the unitarity requirement in the total space and the non-unitary BGK relaxation law in the subsystem. ; After obtaining the effective collision results in the orthogonal moment space, the inverse orthogonal transformation Q-transpose is applied to return to the discrete velocity space, obtaining the updated deviation results after the collision. The overall operator can be expressed as: ; Then, by combining the equilibrium distribution function calculated on classical devices, the post-collision distribution function is reconstructed. It should be noted that the core focus of this invention lies in the quantum implementation method of the BGK collision operator; the migration step and boundary condition step can be handled using existing methods depending on the specific application.
[0031] Compared to schemes that directly construct unified quantum operators in velocity space, this invention first explicitly separates conserved and non-conserved modes, and then uses auxiliary qubits to construct unitary expansions, thus possessing stronger theoretical clarity and physical interpretability. This method is not only applicable to the BGK single-relaxation collision model, but also provides a foundation for subsequent extensions to multi-relaxation time models, moment-space selective control models, and higher-dimensional discrete velocity models.
[0032] Example 1: Simulation of a one-dimensional isothermal density discontinuous flow field
[0033] To verify that the orthogonal moment spatial quantum BGK collision described in this invention can be combined with distribution function migration and macroscopic physical quantity reconstruction to complete the time propagation of a one-dimensional flow field with well-defined initial density and viscosity parameters, this embodiment uses a one-dimensional isothermal density discontinuity Riemannian problem for simulation. This example examines the evolution of the density and velocity fields formed by the combined effects of initial density discontinuity BGK collisions and distribution function migration, without involving the solution of temperature and energy equations. The computational domain is... Divide the region into 400 grid points evenly to calculate the center point of the region. The initial density discontinuity is set with the density on the left side set to 1, the density on the right side set to 0.7, the velocity set to 0, and the kinematic viscosity per lattice unit set to 0.08.
[0034] To verify the flow field simulation effect of the orthogonal moment spatial quantum BGK collision described in this invention, calculations were performed using both the quantum orthogonal moment lattice Boltzmann method and the classical BGK lattice Boltzmann method under the same computational domain, initial density, initial velocity, kinematic viscosity, relaxation rate, discrete velocity model, and boundary conditions. Figure 4 As shown, where Figure 4 (a) shows the density distribution along the computational region after 300 time steps. Figure 4 (b) is the corresponding Directional velocity distribution. Simulation results show that the initial density discontinuity evolves into a high-density region on the left, a density plateau in the middle, and a low-density region on the right under the combined effects of collision and migration. Two transition layers with finite widths are formed between these three regions.
[0035] The density distribution obtained by the method described in this invention largely coincides with the density distribution obtained by the classical lattice Boltzmann method. The transition layer positions and widths obtained by both methods remain consistent, and there are no significant deviations in the density values of the high-density plateau on the left, the middle density plateau, and the low-density plateau on the right. These results demonstrate that the initial fluid density, initial velocity, and kinematic viscosity, after local equilibrium state construction, orthogonal moment space quantum BGK collisions, distribution function migration, and macroscopic physical quantity reconstruction, can be transformed into a density field and a velocity field that evolve over time. The quantum BGK collisions, while maintaining the conserved modes corresponding to mass and momentum, update the non-conserved modes according to the relaxation rate determined by kinematic viscosity, thereby obtaining physical simulation results consistent with the evolution of the classical BGK lattice Boltzmann flow field.
[0036] Example 2: Simulation of a one-dimensional isothermal density discontinuous flow field
[0037] To further illustrate the invention's ability to handle two-dimensional fluid flow including internal solid boundaries, this embodiment employs a two-dimensional periodic domain square pillar scattering problem for simulation. This example places a square solid obstacle within the computational domain. Through orthogonal moment spatial quantum BGK collisions, distribution function migration, and discrete velocity reflection from the square pillar surface, the velocity and vorticity fields of the fluid after bypassing the pillar are obtained. This verifies the invention's ability to simulate internal solid boundaries, wake velocity deficits, and shear layer structures. A two-dimensional computational domain is set up. The computational domain is uniformly divided into 256 and 128 grid points along the x and y directions, respectively. Periodic boundary conditions are applied to both the flow-oriented and lateral outer boundaries of the computational domain. A specular reflection solid boundary condition is applied to the surface of the square pillar, reflecting the distribution function to a discrete velocity direction opposite to the normal velocity component of the square pillar surface while maintaining the tangential velocity component. The incompressible Navier-Stokes reference solution used for comparison employs the same periodic condition at its outer boundary and a corresponding slip-type boundary condition on the square pillar surface to maintain consistent comparison conditions. The fluid density is set to 1, the initial velocity outside the square pillar is 0.03, and the kinematic viscosity per unit grid cell is set to 0.1.
[0038] To verify the simulation effect of the method described in this invention on flows involving internal solid boundaries, the results obtained by this invention were compared with the numerical solution of the incompressible Navier-Stokes equations under the same computational domain, number of meshes, initial velocity, prism size, outer boundary conditions, and specular reflection conditions of the prism. Figure 5As shown above, the initial velocity, viscosity parameters, prismatic geometry, and boundary conditions of the fluid to be simulated can be transformed into a two-dimensional velocity field and vorticity field with clear physical meaning through local equilibrium state construction, orthogonal moment spatial quantum BGK collisions, distribution function migration, and velocity direction reflection. This invention not only obtains the mathematical mapping results of local BGK collisions but also simulates the wake velocity deficit and shear layer structure generated after the fluid interacts with the internal solid boundary.
[0039] The embodiments described above can be further combined or replaced, and these embodiments are merely descriptions of preferred embodiments of the present invention, not limitations on the concept and scope of the present invention. Various changes and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the inventive concept are all within the protection scope of the present invention. The protection scope of the present invention is given by the appended claims and any equivalents.
Claims
1. A method for simulating lattice Boltzmann flow fields based on orthogonal moment spatial quantum BGK collisions, characterized in that, Executed collaboratively by the classical computing module and the quantum computing module, the process includes the following steps: Step 1: Initialize the flow field state: Discretize the fluid region to be simulated using grid points, select the lattice Boltzmann discrete velocity model, set the initial density, initial velocity, viscosity parameters and boundary conditions of the fluid, and initialize the distribution function of each grid point in different discrete velocity directions; Step 2, Calculation of macroscopic fluid parameters and equilibrium state: Based on the zero-order moment and first-order moment of the distribution function of each grid point in the current time step, determine the fluid density and fluid momentum of each grid point, and determine the fluid velocity and local equilibrium state distribution function based on the fluid density and fluid momentum; determine the BGK collision relaxation rate based on the viscosity parameter of the fluid; Step 3, Non-equilibrium distribution construction and quantum state encoding: Calculate the difference between the distribution function and the local equilibrium state distribution function to obtain the non-equilibrium deviation quantity characterizing the non-equilibrium state of the fluid, and encode the non-equilibrium deviation quantity into the system register of the quantum computing module, and initialize the auxiliary qubit to the ground state; Step 4, Orthogonal Moment Space Transformation: Perform orthogonal moment transformation on the system register to map the non-equilibrium deviation from the discrete velocity space to the orthogonal moment space, and divide the modes in the orthogonal moment space into conserved modes corresponding to fluid mass and momentum, and non-conserved modes corresponding to fluid viscous stress and higher-order non-equilibrium information. Step 5, Quantum BGK Collision: Keeping the conserved mode unchanged, determine the rotation parameters of the controlled rotation operation according to the BGK collision relaxation rate, and apply controlled rotation to the auxiliary qubit for the non-conserved mode so that the preserved amplitude of the non-conserved mode when the auxiliary qubit is in the ground state branch is consistent with the relaxation factor corresponding to the BGK collision. Step 6: Collision result extraction and distribution function reconstruction: Measure the auxiliary qubit, extract the system register state corresponding to the auxiliary qubit in the ground state, perform inverse orthogonal moment transformation on the system register state to obtain the non-equilibrium deviation after the collision, and combine the non-equilibrium deviation after the collision with the local equilibrium state distribution function to reconstruct the post-collision distribution function of each grid point; Step 7, Migration and Boundary Processing: Based on each discrete velocity direction, the post-collision distribution function is migrated to adjacent grid points, and the distribution function at the boundary of the computational domain is processed according to the boundary conditions to obtain the distribution function of each grid point in the next time step; Step 8: Macroscopic flow field update: Determine the fluid density and fluid velocity of the next time step based on the distribution function of each grid point in the next time step, repeat steps S2 to S8 until the preset simulation time is reached, and output the density field and velocity field of the fluid region to be simulated.
2. The lattice Boltzmann flow field simulation method according to claim 1, characterized in that, The discrete velocity model is the D2Q9 model, and the orthogonal moment transformation is formed by normalizing orthogonal polynomials on the discrete velocity set into an orthogonal matrix. The orthogonal matrix satisfies unitarity. The conserved modes include density modes, a first-direction momentum mode, and a second-direction momentum mode; the non-conserved modes are determined according to relaxation factors. The contraction occurs, where... Let be the BGK collision relaxation rate.