Quantum implementation of orthogonal moment space BGK collision operator

Abstract

The application discloses a quantum implementation method of a BGK collision operator in an orthogonal moment space, maps a distribution function deviation in a discrete velocity space to an orthogonal moment space, and explicitly separates a conservative mode from a non-conservative mode; an extended unitary evolution participated by an auxiliary quantum bit is constructed for the non-conservative mode, linear relaxation is converted into an implementable unitary process in a total Hilbert space, and an effective mapping equivalent to a classical BGK collision is extracted from an auxiliary bit holding branch; and finally, the velocity space is returned through an inverse orthogonal moment transformation, and collision update is completed. The method establishes a BGK collision implementation path with clear spectral structure, clear physical meaning and convenient quantum circuit expression, compared with directly constructing a black-box quantum operator in a velocity space, the method can keep the conservative mode unchanged and perform an explainable selective relaxation on the non-conservative mode, and provides a unified basis for subsequent expansion of an MRT model, a multi-mode relaxation model and a lattice Boltzmann quantum algorithm.

Description

Technical Field

[0001] This invention relates to the fields of quantum computing and computational fluid dynamics, and more specifically, to an orthogonal moment space quantum realization method for the BGK collision step in the lattice Boltzmann method. 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] Current technologies for handling BGK collisions typically remain at the level of classical velocity space updates, lacking a method that can explicitly distinguish between conserved and non-conserved modes and can structurally implement collision relaxation in quantum systems. In particular, for the non-unitary contraction mechanism corresponding to BGK collisions, a unified construction framework that balances physical interpretability and quantum implementation feasibility is lacking. From a quantum computing perspective, the BGK collision process inherently involves dissipation and contraction characteristics, belonging to non-unitary mappings, and cannot be directly and rigorously implemented in system space by standard quantum gate circuits. Existing research, if it directly constructs quantum operators in velocity space, often struggles to simultaneously maintain physical interpretability, mode-conserving structures, and quantum circuit realizability, especially failing to clearly distinguish the different roles of conserved and non-conserved modes in collisions. On the other hand, the physical essence of LBM collision operators is not an arbitrary linear transformation, but rather the invariance of conserved quantities and the selective relaxation of non-conserved quantities. If this spectral structure cannot be explicitly revealed, the constructed quantum realization path is prone to degenerate into a black box expression lacking physical meaning, which is not conducive to subsequent generalization to more general multi-relaxation models. Therefore, it is necessary to propose a method that can reveal the BGK collision mechanism at the modal level and transform the classical non-unitary relaxation process into a quantum realizable unitary expansion process, in order to solve the problem that the BGK collision operator is difficult to express directly in quantum terms. Summary of the Invention

[0006] To address the difficulty of directly implementing the BGK collision step in existing technologies using quantum mechanics, this invention provides a quantum implementation method for the BGK collision operator in orthogonal moment space. This method does not perform overall black-box quantization of the entire lattice Boltzmann time progression, but instead establishes a structured quantum expression specifically for the BGK collision step.

[0007] The present invention adopts the following technical solution: A quantum implementation method for an orthogonal moment space BGK collision operator, applied to lattice Boltzmann fluid simulation in computational fluid dynamics, includes the following steps: Step 1: Construct the distribution function vector and its equilibrium distribution function vector under the lattice Boltzmann discrete velocity model; calculate the difference between the distribution function and the equilibrium distribution function, which is defined as the deviation. Step 2: Establish an orthogonal transformation from discrete velocity space to orthogonal moment space, map the deviation into modal representation in orthogonal moment space, and divide the modes into conserved modes and non-conserved modes; Step 3: Keep the conserved modes unchanged, introduce auxiliary qubits for the non-conserved modes and construct extended unitary evolution. By applying controlled rotation operations in the composite space composed of the system modes and auxiliary qubits, the non-conserved modes achieve amplitude contraction consistent with the BGK linear relaxation law in the effective subspace where the auxiliary qubits are in the ground state. Step 4: Extract the effective mapping results on the auxiliary qubit ground state preservation branch, and return to the discrete velocity space through inverse orthogonal transformation to obtain the updated results after the BGK collision.

[0008] 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.

[0009] 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.

[0010] 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.

[0011] The effective mapping is obtained by performing post-selection or equivalent bias 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.

[0012] 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.

[0013] 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; 4. This method focuses only on the collision process and can be decoupled and combined with different migration and boundary handling methods, which facilitates the formation of a more complete quantum lattice Boltzmann algorithm. Attached Figure Description

[0014] 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.

[0015] 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.

[0016] 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. Detailed Implementation

[0017] 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.

[0018] 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. f 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 oh Indicates the relaxation rate. f i express i The distribution function of direction, express i The equilibrium distribution function of the direction. This indicates the deviation (unbalance).

[0019] 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.

[0020] Using d'Humieres' linear relaxation model, an invertible unitary matrix of basis transformation is constructed using orthogonal polynomials on discrete velocities. U Q After acting on the initial state, we have: ; The modes are divided into conserved modes. and non-conservative modes .

[0021] As shown in Figure 1, further, an orthogonal matrix is ​​constructed in the discrete velocity space. U Q And map the deviation to the orthogonal moment space: ; because U Q Since it is an orthogonal matrix, its inverse transformation can be obtained by... U Q 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 eigenvalue 1 - . oh and according to the relaxation rate oh Linear contraction occurs: .

[0022] Since the contraction mapping is not a unitary operator in the system space, such as Figure 2As shown in Figure 3, this invention introduces an auxiliary qubit for each non-conservative mode and applies controlled rotations in 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 i This makes cosθ correspond to the contraction coefficient of the non-conservative mode, i.e., cos i = 1 - oh 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 upper part correspond precisely to the modal contraction results required for classical BGK collisions. Here, the revolving door parameters... α and i The relationship satisfies 2 α = i (based on R y (Definition of a door).

[0023] 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...) S : ; Operator S Momentary bases acting on the construction: ; Rotation angle here i With relaxation rate oh Satisfying relation: cos i = 1 - oh .

[0024] 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: ;

[0025] 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.

[0026] 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.

[0027] 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 quantum implementation method for an orthogonal moment space BGK collision operator, applied to lattice Boltzmann fluid simulation in computational fluid dynamics, characterized in that... Includes the following steps: Step 1: Construct the distribution function vector and its equilibrium distribution function vector under the lattice Boltzmann discrete velocity model; calculate the difference between the distribution function and the equilibrium distribution function, which is defined as the deviation. Step 2: Establish an orthogonal transformation from discrete velocity space to orthogonal moment space, map the deviation into modal representation in orthogonal moment space, and divide the modes into conserved modes and non-conserved modes; Step 3: Keep the conserved modes unchanged, introduce auxiliary qubits for the non-conserved modes and construct extended unitary evolution. By applying controlled rotation operations in the composite space composed of the system modes and auxiliary qubits, the non-conserved modes achieve amplitude contraction consistent with the BGK linear relaxation law in the effective subspace where the auxiliary qubits are in the ground state. Step 4: Extract the effective mapping results on the auxiliary qubit ground state preservation branch, and return to the discrete velocity space through inverse orthogonal transformation to obtain the updated results after the BGK collision.

2. The quantum realization method according to claim 1, characterized in that, 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.

3. The quantum realization method according to claim 1, characterized in that, 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.

4. The quantum realization method according to claim 1, characterized in that, 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.

5. The quantum realization method according to claim 1, characterized in that, The effective mapping is obtained by performing post-selection or equivalent bias 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.

6. The quantum realization method according to claim 1, characterized in that, The method establishes a quantum realization process for the BGK collision step in the lattice Boltzmann method, excluding the quantum encoding of the transfer step and the boundary condition step; The calculation of the equilibrium distribution function and the reconstruction of the distribution function after the collision can be performed on classical computing devices.

Images