Computing method of scalar conservation law equation, computing device, quantum computing system, and storage medium

Abstract

A kind of calculation method of scalar conservation law equation, computing device, quantum computing system and storage medium, can overcome the dimension disaster existing when high-dimensional solution, so as to reduce the cost of calculation, improve the efficiency of calculation.The calculation method comprises: by level set dimension increasing transformation, scalar conservation law equation is converted into linear Liouville equation;Linear Liouville equation is spatially dispersed by finite difference method, and is converted into non-hermitian matrix system;Non-hermitian matrix system is processed by Schrödinger, and the hermitian Hamilton system with hermitian Hamilton is constructed;Based on the hermitian Hamilton, construct continuous controlled evolution operator, and convert continuous controlled evolution operator into approximate controlled evolution operator;Based on approximate controlled evolution operator, construct quantum circuit;Prepare the initial quantum state of quantum circuit, execute quantum circuit, obtain the output quantum state of quantum circuit;The output quantum state is measured, and the numerical solution of scalar conservation law equation is determined based on the measurement result.

Description

Technical Field

[0001] This disclosure relates to the field of quantum computing, specifically to a method for calculating scalar conservation law equations, a computing device, a quantum computing system, and a storage medium. Background Technology

[0002] As is well known, numerical computation is an important tool for the evolution analysis of physical systems and the simulation of engineering applications. On the other hand, scalar conservation law equations, as typical nonlinear hyperbolic partial differential equations describing the transport process of conserved quantities, are widely used in many technical fields such as fluid mechanics, transportation engineering, oil and gas reservoir development, and supply chain logistics. These equations can accurately characterize key physical phenomena such as shock wave formation, discontinuous solution evolution, and density wave propagation, and are important mathematical tools for dynamic process modeling and numerical prediction.

[0003] Traditional numerical methods for solving scalar conservation law equations mainly include the finite volume method, the discontinuous Galerkin method, and the weighted essentially non-oscillatory (WENO) scheme. All of these methods rely on grid discretization to achieve numerical iterative solutions to partial differential equations. In low-dimensional scenarios, these classical numerical methods can obtain relatively stable and reliable computational results. However, as the dimensionality of the problem space increases, the number of discrete grid points grows exponentially with the dimensionality, leading to a sharp increase in computational and storage costs, resulting in the severe "curse of dimensionality." Furthermore, due to the stability constraints of explicit iterative schemes, the time step must be reduced synchronously with the spatial grid size, further exacerbating the overall computational complexity and making it difficult to achieve efficient and high-precision solutions to high-dimensional scalar conservation law equations.

[0004] In recent years, quantum computing, with its quantum superposition, quantum entanglement, and quantum parallelism properties, has provided a new technical path to overcome the performance bottleneck of classical computing, demonstrating significant potential speedup advantages in scenarios such as solving linear equations, Fourier transforms, and Hamiltonian simulations. However, directly applying quantum computing to solving scalar conservation law equations still faces insurmountable technical obstacles: on the one hand, scalar conservation law equations have strong nonlinear characteristics, while general quantum algorithms can only directly handle linear dynamic processes, and nonlinear constraints make quantum simulations difficult to apply directly; on the other hand, the system matrix obtained after discretization is usually a non-Hermitian matrix, which cannot directly construct unitary operators that satisfy the requirements of quantum evolution, making it difficult to transform classical dynamics into legitimate quantum evolution processes.

[0005] Furthermore, existing quantum solution frameworks generally lack a complete linearization and quantization pathway for nonlinear conservation laws, making it difficult to efficiently transform nonlinear partial differential equations into executable quantum circuits. Even when quantum simulations of some components are achieved through approximation, problems such as complex quantum circuit structures, high gate overhead, high measurement complexity, and cumbersome numerical solution extraction processes are common, making it impossible to achieve stable, efficient, and scalable quantum solutions for scalar conservation law equations while ensuring physical conservation properties and solution accuracy. Summary of the Invention

[0006] This disclosure is made in consideration of the above circumstances, and its purpose is to provide a scalar conservation law calculation method that can overcome the curse of dimensionality in high-dimensional solutions, transform nonlinear non-Hermitian systems into quantum-simulatable unitary evolutions, and achieve efficient, stable, and low-complexity numerical solutions, thereby reducing computational costs and improving computational efficiency.

[0007] One technical solution disclosed herein provides a method for calculating scalar conservation law equations, including: The first step is to transform the scalar conservation law equation into a linear Liouville equation through a level set dimension-up transformation. The second step is to spatially discretize the linear Liouville equation using the finite difference method, transforming it into a non-Hermitian matrix system. The third step is to perform Schrödingerization on the non-Hermitian matrix system to construct a Hermitian Hamiltonian system with Hermitian Hamiltonian. The fourth step is to construct a continuous controlled evolution operator based on the Hermi-Hamiltonian, and then transform the continuous controlled evolution operator into an approximate controlled evolution operator through Li-Trott-Suzuki decomposition. The fifth step involves constructing a quantum circuit that can run on a quantum processor or quantum simulator based on the approximate controlled evolution operator, the preset QFT unit, the Oracle operator, the conjugate Oracle operator, and the IQFT unit. The sixth step is to prepare the initial quantum state of the quantum circuit and execute the quantum circuit based on the initial quantum state to obtain the output quantum state of the quantum circuit. The seventh step is to measure the output quantum state and determine the numerical solution of the scalar conservation law equation based on the measurement results.

[0008] According to the calculation method described in this technical solution, the nonlinear non-Hermitian scalar conservation law equation can be transformed into a unitary evolution process that can be directly executed on quantum hardware. This effectively overcomes the curse of dimensionality in classical numerical methods, significantly reduces the computational complexity and storage overhead of high-dimensional numerical solutions, and improves solution stability and numerical accuracy. Attached Figure Description

[0009] Figure 1 This is a flowchart illustrating the main steps of the Schrödingerization process.

[0010] Figure 2 This is a schematic diagram of a quantum circuit representing an auxiliary unitary operator.

[0011] Figure 3 This is a flowchart illustrating a method for calculating the scalar conservation law equation according to an embodiment of this disclosure.

[0012] Figure 4 This is a schematic diagram of a quantum circuit representing the first many-body global flip evolution term.

[0013] Figure 5 This is a schematic diagram of a quantum circuit representing a unitary transformation.

[0014] Figure 6 This is a schematic diagram of a quantum circuit representing the first single-unit evolution term.

[0015] Figure 7 This is a schematic diagram of a quantum circuit representing the first approximate basic controlled evolution operator.

[0016] Figure 8 This is a schematic diagram of a quantum circuit representing the second monomer rise and fall evolution term.

[0017] Figure 9 This is a schematic diagram of a quantum circuit representing the second approximate basic controlled evolution operator.

[0018] Figure 10 This is a schematic diagram of a quantum circuit representing a window evolution operator.

[0019] Figure 11 This is a schematic diagram of a quantum circuit representing an approximate controlled evolution operator for a single time step.

[0020] Figure 12 This is a schematic diagram of a quantum circuit used to solve the scalar conservation law equation.

[0021] Figure 13 This is a schematic diagram of a quantum circuit representing the measurement scheme used in the calculation method described in an embodiment of this disclosure.

[0022] Figure 14 This is a schematic diagram illustrating the main architecture of a quantum computing system according to an embodiment of the present disclosure.

[0023] Figure 15 This is a schematic diagram illustrating the hardware architecture of a computing device according to an embodiment of the present disclosure.

[0024] Figure 16This is a graph showing the fitting curves of the numerical solutions obtained by solving the one-dimensional inviscid Burgers equation, an example of a scalar conservation law equation, using the calculation method described in an embodiment of this disclosure, the classical Schrödinger method, and the existing finite difference matrix multiplication method, based on different auxiliary qubit numbers.

[0025] Figure 17 This represents the fitting curves of the numerical solutions obtained by solving a one-dimensional classical traffic flow model, which is another example of the scalar conservation law equation, using the calculation method described in an embodiment of this disclosure, the classical Schrödinger method, and the existing finite difference matrix product method, based on different auxiliary qubit numbers.

[0026] Figure 18 This is a schematic diagram showing the analysis results obtained by analyzing the quantum gate complexity when solving the one-dimensional inviscid Burgers equation, which is an example of a scalar conservation law equation, using the computational method described in an embodiment of this disclosure.

[0027] Figure 19 This is a graph showing the relationship between the second-order norm error and the number of auxiliary qubits in the numerical solution of the one-dimensional inviscid Burgers equation, which is an example of a scalar conservation law equation, when the computational method described in an embodiment of this disclosure is used. Detailed Implementation

[0028] (Description of the object being processed)

[0029] The following is an overview of the scalar conservation law partial differential equations, which are the objects processed by the calculation method described in this disclosure.

[0030] Specifically, the computational method described in this disclosure handles the Cauchy problem of scalar conservation laws through numerical computation, and this problem can be expressed in the following form:

[0031] in: d represents the spatial dimension; u(x,t) is the conserved scalar field to be solved, which represents the conserved quantity of the physical system as it changes with time and space. x is the spatial variable and t is the time variable. F(u) is the flux function, which characterizes the transport rate of the above-mentioned conserved quantities; u0(x) represents the initial conditions of the above problem and defines the spatial distribution of the conserved quantities at time t=0.

[0032] The scalar conservation law equations described here are mathematical models of transport phenomena in physical systems, widely used in fluid mechanics, traffic engineering, and oil and gas reservoir engineering. For example, in traffic engineering, the Light-Whitham-Richards (LWR) model typically uses scalar conservation law equations to describe the evolution of vehicle density, analyzing the formation and propagation of traffic congestion. As another example, in oil and gas reservoir engineering, the Buckley-Leverett equations, as a specific example of scalar conservation law equations, are often used to describe the motion of oil-water two-phase flow in porous media.

[0033] As described in the background section, the mathematical description of the above equations reveals that this type of partial differential equation exhibits nonlinear characteristics. Consequently, even if the initial condition u0(x) possesses smooth properties, the nonlinearity of the flux function will cause the characteristic lines to intersect within a finite time interval, resulting in discontinuous solutions such as shock waves. Therefore, the solution needs to be calculated within a weak solution framework. The calculation method described in this disclosure is based on this premise.

[0034] (Description of some technical terms)

[0035] Before describing specific embodiments, some technical terms used in this disclosure will be explained to facilitate understanding and clarify the scope of protection.

[0036] The scalar conservation law equations described in this disclosure are nonlinear hyperbolic partial differential equations that describe the transport and evolution of conserved quantities in physical systems. They are widely used in numerical calculations of shock waves, discontinuous solutions, and transport processes in scenarios such as fluid mechanics, transportation engineering, and oil and gas reservoir engineering.

[0037] The level set described in this disclosure refers to the set of all points that satisfy the condition that a certain function takes the value of a constant. In this disclosure, it specifically refers to the solution set determined by the level set function taking the value of zero, which is used to characterize the solution surface of the scalar conservation law.

[0038] The level set dimensionality-upgrading transformation described in this disclosure refers to a transformation method that transforms nonlinear scalar conservation law equations into high-dimensional linear transport equations by introducing additional solution space variables and constructing level set functions. In this disclosure, it is used to achieve equation linearization and avoid non-unique solutions and shock wave oscillation problems caused by nonlinearity.

[0039] The linear Liouville equation described in this disclosure refers to the linear transport partial differential equation obtained after the above-mentioned level set dimensionality-up transformation, which is used to transform the original nonlinear equation into a linear dynamic form suitable for quantum simulation.

[0040] The finite difference method described in this disclosure refers to a numerical discretization method that divides the spatial variables and the solution space variables introduced by the dimensionality increase transformation of the level set into a grid, and uses forward difference, backward difference, upwind scheme, and other methods to transform the continuous partial differential equation into a discrete algebraic equation (matrix equation). This method can ensure that the discretization process satisfies the physical conservation and CFL stability conditions.

[0041] The Hermitian matrix described in this disclosure refers to a class of matrices whose conjugate transpose is equal to itself, whose eigenvalues ​​are real numbers, and which form the mathematical basis for observable physical quantities and quantum unitary evolution in quantum mechanics.

[0042] The Schrödingerisation described in this disclosure is a numerical processing technique that transforms non-unitary classical dynamical systems into high-dimensional unitary evolutions, making them suitable for quantum simulation. By introducing auxiliary variables and performing Fourier transforms on them, non-Hermitian linear ordinary differential equations (ODEs) can be transformed into Schrödinger equations driven by Hermitian Hamiltonians, providing a foundation for subsequent construction of controlled evolution operators and realization of quantum circuits.

[0043] The Hamiltonian described in this disclosure is a class of Hermitian operators that drive the time evolution of quantum states in quantum mechanics. In this disclosure, it is the core mathematical foundation for constructing continuously controlled evolution operators, performing Lie-Trott-Suzuki decomposition, and realizing quantum circuits.

[0044] The continuous controlled evolution operator described in this disclosure refers to a unitary evolution operator generated by a Hermi-Hamiltonian and executed with a fixed time step, characterizing the continuous-time evolution of quantum states.

[0045] The discrete controlled evolution operator (approximate controlled evolution operator) described in this disclosure refers to a discrete evolution operator that can be directly executed on quantum hardware, formed by approximating a continuous controlled evolution operator through Li-Trott-Suzuki decomposition.

[0046] The Li-Trotter-Suzuki Decomposition (LTS decomposition) described in this disclosure is an approximate method for decomposing the exponential evolution of a complex Hamiltonian into a product of multiple simple operators. It can decompose the many-body quantum evolution, which is difficult to realize directly, into a form that can be realized by standard quantum gates.

[0047] The quantum gates described in this disclosure are the basic units in quantum circuits that perform basic unitary operations on qubits, including single-qubit gates, controlled gates, multi-controlled gates, and Pauli gates, and are the basic units that constitute quantum algorithms.

[0048] The quantum register described in this disclosure is a collection of qubits used to carry a quantum state. In this disclosure, it includes an index register, an auxiliary register, a de-register, and a space register, which are used to store grid index, auxiliary parameters, de-quantum state, and space variable quantum state, respectively.

[0049] The quantum Fourier transform (QFT) unit described in this disclosure is a quantum circuit module that implements discrete Fourier transform on a quantum processor or quantum simulator, used to transform quantum states from the spatial domain to the frequency domain to reduce the complexity of quantum evolution.

[0050] The inverse quantum Fourier transform (IQFT) unit described in this disclosure is the inverse operation module of the quantum Fourier transform, used to inversely transform the frequency domain quantum state back to the spatial domain for quantum measurement and numerical solution extraction.

[0051] The Oracle operator (quantum oracle operator) described in this disclosure is a unitary quantum operation that encapsulates classical function mappings and is used to encode classical parameters corresponding to discrete grid indices into an auxiliary quantum register to provide control signals for controlled evolution.

[0052] The conjugate Oracle operator described in this disclosure is the conjugate transpose unitary operation of the aforementioned Oracle operator, used to restore the auxiliary quantum register to its initial state after evolution is completed, thereby ensuring the unitarity and evolution fidelity of the entire quantum circuit.

[0053] The quantum circuit described in this disclosure is a sequence of instructions consisting of quantum registers, quantum gates, oracle operators, conjugate oracle operators, QFT units, IQFT units, and measurement operations, which can run on a quantum processor or quantum simulator to implement a quantum solution algorithm for scalar conservation equations.

[0054] The measurement described in this disclosure refers to the operation of performing computational basis measurement on the output quantum state of a quantum circuit, statistically analyzing the probability distribution of the measurement results, and extracting numerical solutions to the scalar conservation law equations from them.

[0055] The numerical solution described in this disclosure refers to the approximate physical solution of the scalar conservation law equation on a discrete spatial grid, obtained by measuring and post-processing the output quantum state. It can be directly applied to engineering scenarios such as fluid mechanics, transportation, and oil and gas reservoirs.

[0056] (Description of Schrödingerization)

[0057] Before describing the calculation method in detail in the specific embodiments of this disclosure, the basic principles and steps of Schrödingerisation will be explained.

[0058] Figure 1This is a flowchart illustrating the main steps of the Schrödinger transformation. When solving a nonlinear partial differential equation, the first step is usually to linearize the equation, either by reducing its linearity or by using methods such as the finite element method or the finite difference method to transform the solution into a linear system. The solution is obtained. The Schrödinger transformation is applied to this linear system. Furthermore, the solution u of this linear system can be considered as the steady-state solution of the following equation. :

[0059] When the above coefficient matrix When the matrix is ​​a positive semi-definite matrix (which ensures the stability of the linear system), the linear system can be solved by quantum simulation through the Schrödinger transformation process described here.

[0060] First, in step ST001, the above non-homogeneous linear system is rewritten in the following form:

[0061] In other words, in the original solution Auxiliary variables were introduced based on this. Constructing augmented vectors This transforms the aforementioned non-homogeneous linear system into a homogeneous linear system, where:

[0062] Next, in step ST002, the above coefficient matrix is... It is decomposed into a combination of Hermitian and anti-Hermitian matrix terms. Generally speaking, the coefficient matrix of the above homogeneous linear system... It is not a Hermitian matrix and cannot be used for quantum simulation calculations. Therefore, the above coefficient matrix is... Decomposed into a combination of the following Hermitian and anti-Hermitian matrix terms: .

[0063] Then, in step ST003, variables are further introduced. The variable domain is extended through the following warped phase transformation: .

[0064] On the other hand, in step ST004, the above homogeneous linear system is transformed into a Hermi-Hamiltonian system through discrete Fourier transform:

[0065] in, The momentum operator under the original variable The diagonal matrix representation of .

[0066] (Description of the quantum representation of the finite difference method)

[0067] Next, before proceeding with a detailed description of the computation method described in the specific embodiments of this disclosure, the quantum representation of the finite difference method will be introduced to elucidate the theoretical basis and discretization implementation of the computation method described in this disclosure.

[0068] One of the most commonly used numerical methods for solving partial differential equations is the finite difference method. In quantum computing, the differential operators used in the finite difference method can be represented by quantum gates.

[0069] For ease of understanding, we will use a one-dimensional case as an example.

[0070] Consider a one-dimensional region Discretize it into A uniformly distributed grid of points, among which .vector This represents the solution to the equation. Applying this vector to differential operators yields different representations of the differential operators:

[0071] Among them, u N and u -1 The definition of u depends on the boundary conditions. For example, when periodic boundary conditions are used, u N =u0andu -1 =u N-1 .

[0072] On the other hand, the finite difference operator can be represented as a matrix product operator (MPO) using the following three 2×2 matrices:

[0073] in, .

[0074] Based on this, the translation operator is defined. and as follows:

[0075] in, .

[0076] Therefore, the difference operator can be expressed as the raising and lowering operator. and The combination of . Here, for ease of understanding, the mathematical representation of the one-dimensional forward difference operator under periodic boundary conditions is shown as an example: .

[0077] Building upon this, in order to realize the evolutionary computation of quantum circuits, the difference operator needs to be transformed from the static matrix form described above into a dynamic evolutionary form that can be executed on quantum circuits. To this end, an auxiliary Hamiltonian of the following form is constructed for subsequent approximation of the difference operator's function through time evolution:

[0078] in, It's a scaling parameter. It is a phase parameter.

[0079] Therefore, the following continuously controlled evolution operator can be constructed: .

[0080] It is known that current quantum circuits can only realize discrete sequences consisting of a finite number of single-qubit gates and controlled gates. However, the aforementioned controlled evolution operator is a continuous, exponential operator that cannot be realized using a finite number of basic quantum gates. Furthermore, the aforementioned auxiliary Hamiltonian is a complex many-body operator, the sum of multiple raising and lowering operators, which typically cannot be realized simultaneously and require approximation through decomposition. Specifically, since the aforementioned auxiliary Hamiltonian contains multiple raising and lowering operators acting on different qubits, these operators are typically non-commutative and cannot be realized simultaneously using a finite number of quantum gates.

[0081] On the other hand, Belki is defined as It can also usually be expressed as Therefore, the following auxiliary unitary operator (auxiliary unitary operator) can be constructed:

[0082] in, It acts on the first An Adama gate for 1 qubit, It acts on the first Phase gate for 1 qubit, Therefore, the first The target is the 1st qubit. Each qubit is a controlled NOT gate with control bits. Figure 2 A schematic diagram of the quantum circuit for the aforementioned auxiliary unitary operator is shown.

[0083] It should be noted that the auxiliary unitary operator mentioned here is used to adjust the phase of the differential operator, and is used to transform the combination of the above-mentioned raising and lowering operators into a form that can be realized by quantum circuits.

[0084] Furthermore, as mentioned above, the aforementioned continuous controlled evolution operator cannot be directly implemented using quantum circuits. Therefore, the inventors of this disclosure propose performing a Lie-Trott-Suzuki decomposition (hereinafter referred to as LTS decomposition) on the aforementioned continuous controlled evolution operator, and then further transforming it into a standard quantum circuit form using the aforementioned auxiliary unitary operator. Through LTS decomposition and the auxiliary unitary operator, the aforementioned continuous controlled evolution operator can be transformed into the following approximate controlled evolution operator (sometimes also called a discrete controlled evolution operator):

[0085] in, It is a time step. It acts on the first The qubit and with the th qubit Each qubit is a control bit for multiple controlled qubits Door.

[0086] (Description of the calculation method)

[0087] Next, based on the above explanation, we will refer to Figures 3 to 13 The main steps of the calculation method of the scalar conservation law equation according to an embodiment of the present disclosure are described below.

[0088] First, in step ST100 (i.e. the first step), the scalar conservation law partial differential equation with the above initial conditions is subjected to a level set dimension-up transformation to convert the scalar conservation law equation into a linear Liouville equation.

[0089] Specifically, in step ST100, for the scalar conservation law partial differential equation with initial conditions, the level set function is defined as follows:

[0090] in, , Representing dimension, It is an introduced variable that increases the dimensionality of the solution space.

[0091] It should be noted that the above level set function can be explicitly expressed as:

[0092] when hour, Therefore, the solution to the original partial differential equation is... The zero isosurface corresponds to the level set function.

[0093] Substituting the above level set function into the above scalar conservation law partial differential equation and its initial conditions, we can obtain the following mathematical expression:

[0094] Thus, through the above actions, the d+1-dimensional nonlinear partial differential equation can be transformed into a 2d+1-dimensional first-order linear transport equation.

[0095] Furthermore, considering the aforementioned level set function It is an implicit isosurface representation that describes the scalar conserved field. yes The point set, but this form itself is not suitable as the initial condition for a quantum state.

[0096] Therefore, in this embodiment, a distribution function is further introduced. The distribution function is an explicit probability density representation, which in It takes a non-zero value at one position and a zero value at the other positions. This distribution function can encode the amplitude distribution of quantum states. In other words, the level set function... As a geometric isosurface function, it lacks distribution density properties and cannot be directly encoded as a quantum. Therefore, it is necessary to convert it into a distribution function. To adapt to the amplitude encoding and evolution of quantum states.

[0097] With the introduction of the above distribution function, the level set function can be equivalently transformed into a distribution function:

[0098] Therefore, it can be seen that the distribution function is formed by using the Dirac function to "thicken" the zero isosurface of the level set into a distribution.

[0099] Thus, by transforming the level set function into the aforementioned distribution function, the first-order linear transport equation can be converted into a linear Liouville equation:

[0100] For ease of understanding, another formulation of the above linear Liouville equation is shown here:

[0101] Therefore, the initial conditions are correspondingly transformed into the following expression:

[0102] Next, in step ST200 (i.e., the second step), the linear Liouville equation obtained in step ST100 is spatially discretized using the finite difference method, transforming the linear Liouville equation into a time-continuous non-Hermitian matrix system. In other words, the purpose of step ST200 is to transform the above-mentioned linear Liouville equation into a linear algebra problem (i.e., solving a system of linear equations) through discretization.

[0103] Specifically, for spatial variables Reconciliation space variables Spatial grid partitioning is performed, discretizing the continuous variables into a finite number of grid points. Then, discretization schemes such as forward differencing and backward differencing are used to replace the partial derivative operators in the linear Liouville equation with finite difference operators. This transforms the continuous partial differential equation into a discrete distribution function. Matrix form of the differential equation:

[0104] For ease of understanding, the distribution function is discussed here. The specific discretization method (time discretization is also explained here for ease of subsequent explanation) will be explained. Specifically, the distribution function... It can be converted into the following discrete form:

[0105] in: Subscript Indicates the time step, i.e. ; vector , Represent spatial vectors respectively Spatial grid index and solution space vector The spatial grid index, corresponding to the grid size is , Grid points are defined as , ,in , , .

[0106] However, considering that the initial conditions include the Dirac function, a smooth approximation function is required. It is approximated as follows:

[0107] The above smooth approximation function has the following properties: when hour, ; Satisfying the normalization condition, that is, satisfying .

[0108] Therefore, the specific discrete forms of the time derivative term and the convection term in the above linear Liouville equation are as follows: Discrete time derivative term

[0109] Convection term discretization

[0110] in: , ; , It is the translation operator in the direction of the spatial vector x, defined as

[0111] Right now, This indicates that the current spatial grid will be... The function value at that point is shifted to its adjacent grid point on the right. The right translation operator at the location, This indicates that the current spatial grid will be... The function value at that point is shifted to its adjacent grid point on the left. The left translation operator at the location.

[0112] It can be observed that, With spatial variables Time variables Since it is irrelevant, it can be treated as a constant when solving linear systems.

[0113] Furthermore, to ensure the stability of the equations, an upwind scheme is employed in this embodiment. Specifically, the definition is as follows: And the following CFL conditions must be met:

[0114] Furthermore, based on the above discretization, it can be seen that in the discretized equations described above, It is only related to time, therefore it can be regarded as a variable that is only related to time. In other words, the difference operator acts on The above can be represented as and .

[0115] To avoid loss of generality, assume that when hour, , and when hour, .

[0116] Therefore, by rearranging the above matrix form differential equation, we can obtain:

[0117] Furthermore, regarding the solution space variables Discretize. Specifically, let The above interval is discretized into multiple Each Describes the solution space variables The The first dimension Each grid index. It is important to note that this is to cover the initial conditions, i.e., the initial solution. Let P satisfy the range of values ​​of . .

[0118] To simplify the analysis, we will only consider... i.e., solution space variables The case where the dimension is one-dimensional (therefore, it will be omitted below) subscript The situation is similar for multidimensional cases.

[0119] In addition, to simplify the notation, record and ,in, Describing the flux derivative In solving space variables The The values ​​at each grid point.

[0120] In this way, the above partial differential equation in matrix form can be further transformed into the following ordinary differential equation in matrix form:

[0121] in, This is the discretized coefficient matrix (system matrix). Generally speaking, the above system matrix... It is not a Hermitian matrix and requires further processing to be applicable to quantum computing (see step ST300 for details).

[0122] In this way, the transformation from the linear Liouville equation to a discretized non-Hermitian matrix system is completed.

[0123] Then, in step ST300 (i.e. the third step), the non-Hermitian matrix system obtained in step ST200 is subjected to Schrödinger transformation to construct a Hermitian Hamiltonian system with Hermitian Hamiltonian quantities.

[0124] First, as mentioned above, the coefficient matrix of the above ordinary differential equations is generally... It is not a Hermitian matrix. Therefore, in this step, we first convert the above coefficient matrix... Decomposed into two Hermitian matrices, namely the first Hermitian matrix. Second Hermitian matrix The combination is :

[0125] Next, auxiliary variables are introduced using the twisted phase transform described above. And construct augmented variables (or augmented state vectors). It can also be represented in the following vector form:

[0126] Therefore, the above non-Hermitian matrix system can be further transformed into the following intermediate evolutionary matrix system:

[0127] At this point, the initial conditions (or initial data) are... .

[0128] Then, regarding auxiliary variables Using Fourier transform, the above Hermitian matrix system is further transformed into a Hermitian Hamiltonian system with Hermitian Hamiltonians:

[0129] in, ( ) is an auxiliary variable Fourier transform conjugate variables, It is the first Hermitian matrix mentioned above. Second Hermitian matrix A linear combination of these is also a Hermitian matrix. Furthermore, the above form... This can serve as a Hamiltonian in quantum simulations; therefore, it is referred to here as the Hermi-Hamiltonian. Based on this, the aforementioned matrix system is a time-evolutionary system driven entirely by this Hermi-Hamiltonian, hence it can be called a Hermi-Hamiltonian system. Furthermore, it can be observed that the form of this equation is similar to the standard Schrödinger equation. It has the same form and can be used for unitary evolution in quantum simulation.

[0130] In addition, in order to perform unitary evolution, auxiliary variables need to be adjusted. Discretization is performed. Specifically, let... The grid size is Furthermore, define ,in .

[0131] Therefore, the initial conditions can be expressed as Its Fourier transform is .set up as auxiliary variables The corresponding number of qubits satisfies .

[0132] Here, taking the satisfying of periodic boundary conditions as an example, we examine the Hermi-Hamiltonian. The detailed expansion form will be explained below. First, the solution space vectors are listed here. The Hermi-Hamiltonian is a one-dimensional system that satisfies periodic boundary conditions. Detailed expansion format:

[0133] in: express Tensor product of identity matrices; , indicating the first indivual Flux derivative at discrete points The value; Forward difference operator , It is the right translation operator mentioned above; Backward difference operator , It is the right translation operator mentioned above; Central difference operator ; Symmetric difference operator .

[0134] Furthermore, the forward difference operator and the backward difference operator can also be expressed as follows:

[0135] in addition, and These are referred to as the central difference Hamiltonian component and the symmetric Hamiltonian component, respectively. Both are Hermitian matrices and are described as follows:

[0136] To simplify the expression, let's assume... as well as .

[0137] Then, in step ST400 (i.e. the fourth step), a continuous controlled evolution operator with a single time step is constructed based on the above Hermi-Hamiltonian, and the continuous controlled evolution operator is transformed into a discrete controlled evolution operator with a single time step through LST decomposition.

[0138] Specifically, for a single time step Based on the above Hermi-Hamiltonian The following continuously controlled evolution operators with a single time step can be constructed. :

[0139] Next, the LST decomposition is used to decompose the above-mentioned continuous controlled evolution operator, so that the above-mentioned continuous controlled evolution operator with a single time step can be approximately transformed into the following discrete controlled evolution operator with a single time step. :

[0140] in, and Based on the above central difference Hamiltonian components and symmetric Hamiltonian components Two fundamental controlled evolution operators are constructed (referred to as the first fundamental controlled evolution operator and the second fundamental controlled evolution operator, respectively). However, these fundamental controlled evolution operators are abstract mathematical expressions and need to be transformed into concrete quantum circuits composed of standard quantum gates that can be directly implemented on a quantum computer or quantum virtual machine (or quantum simulator). Therefore, the following sections will discuss... and To perform the conversion of specific quantum circuits.

[0141] First, consider the first fundamental controlled evolution operator. The specific transformation of quantum circuits.

[0142] Consider the following equation:

[0143] in,

[0144] The first basic controlled evolution operator described above can be further transformed into the following mathematical expression:

[0145] According to the above mathematical formula, the first fundamental controlled evolution operator... It can be broken down into the first global phase correction term. First multi-body global flip evolution term and the first single-unit rise and fall evolution term .

[0146] First global phase correction term It is a constant phase factor separated from the exponent of the first basic controlled evolution operator mentioned above, used for all... Each qubit is subjected to the same phase shift Operation. The mathematical expression of the first global phase correction term is as follows:

[0147] According to this mathematical expression, the term consists of two parts: a scalar exponential factor. tensor product of multi-bit identity matrix .

[0148] scalar exponential factor It is the global complex phase factor of the entire system, characterizing the effect of... A unified overall phase shift is applied to the quantum state of the entire system composed of qubits. At the same time, it does not change the probability distribution of each calculated ground state.

[0149] Multi-bit identity matrix tensor product Indicates all Each qubit undergoes a unit operation, and the qubit state remains unchanged. In other words, this tensor product represents all... All qubits are simultaneously in a "no operation" state.

[0150] Regarding the hardware implementation of the aforementioned first global phase correction term, in the case of superconductivity, virtual Z-rotation is preferred. With virtual Z-rotation, by shifting the reference frame phase of subsequent microwave control pulses, a phase shift of the entire system can be effectively introduced without applying physical pulses to the quantum bits. .

[0151] First multi-body global flip item This describes the many-body interaction in which all qubits in the system simultaneously undergo either a 0→1 or a 1→0 flip. Under periodic boundary conditions, this term is used to implement the boundary cycle term in the difference operator to ensure the numerical stability of the difference scheme. Figure 4 A schematic diagram of the quantum circuit corresponding to the first many-body global flip term is shown. The quantum circuit expression of the first many-body global flip term is as follows:

[0152] in: Indicates the action in the The unitary transformation on each qubit is a control sequence consisting of Adama gates and CNOT gates, used to transpose the basis vectors required for the controlled operation of the standard computation basis vectors (the specific mathematical expression of which has been mentioned above and will not be repeated here). Figure 5 A schematic diagram of the quantum circuit corresponding to this unitary transformation is shown; yes The inverse operation of the two forms a conjugate in the above terms, the purpose of which is to transform the operation between the two into a new basis vector for execution. It is Pauli The gate is a single-qubit unitary gate used to flip the state of the target bit (i.e., , In particular, in the above items, Appearing in pairs The front and back of the door are used to control Revolving door Converted into a controlled phase gate to achieve Multibody flipping; It is a multi-control bit controlled Z-rotation door, with the control bits being the front bit. The target bit is the nth qubit. A number of qubits, only when all the preceding bits are used as control bits Each quantum bit is in Only with this attitude can one address the first... Each qubit is subjected to a rotation angle about the Z-axis. .

[0153] First single-unit ascending and descending evolution term Characterizes the independent rise and fall evolution of each qubit, used to realize the central difference approximation of the spatial derivative. Figure 6 A schematic diagram of the quantum circuit for the first monomer rise-fall evolution term is shown. The quantum circuit expression for this first monomer rise-fall evolution term is as follows:

[0154] Since the meanings of each part in the first single-unit evolution term have already been explained above, a repeated description is omitted here.

[0155] Therefore, the first basic controlled evolution operator described above can be approximately transformed into the following first approximate basic controlled evolution operator:

[0156] Figure 7 A schematic diagram of the quantum circuit corresponding to the first approximate basic controlled evolution operator is shown.

[0157] Next, consider the second fundamental controlled evolution operator. The specific transformation of quantum circuits.

[0158] Second fundamental controlled evolution operator This can be further expressed as:

[0159] According to the above mathematical formula, the second basic controlled evolution operator... It can be decomposed into a second many-body global flip evolution term. and the second monomer rise and fall evolution term .

[0160] Second multi-body global flip evolution term Describes all This involves multiple qubits simultaneously participating in a many-body interaction with antisymmetric phase characteristics. Under periodic boundary conditions, this second many-body global flip evolution term is used to realize the antisymmetric phase evolution of the boundary cycle term in the difference operator, ensuring the stability and symmetry of the numerical scheme. That is, this second many-body global flip evolution term corresponds to the first many-body global flip evolution term mentioned above, but the phase evolution direction is opposite. Here, the quantum circuit expression of this second many-body global flip evolution term is shown:

[0161] Second single-unit ascending and descending evolution term The antisymmetric rise and fall evolution, representing the independent evolution of each qubit, is the antisymmetric component used to realize the center-difference approximation of the spatial derivative. That is, this second single-qubit rise and fall evolution term corresponds to the first single-qubit rise and fall evolution term mentioned above, but its evolution form is... . Figure 8 A schematic diagram of the quantum circuit for the second monomer rise-fall evolution term is shown. Here, the quantum circuit expression of this second monomer rise-fall evolution term is shown:

[0162] Therefore, the above-mentioned second basic controlled evolution operator can be approximately transformed into the following second approximate basic controlled evolution operator:

[0163] Figure 9 A schematic diagram of the quantum circuit corresponding to the second approximate basic controlled evolution operator is shown.

[0164] Based on the above analysis, the continuous controlled evolution operator It can be further approximated as:

[0165] However, it should be noted that here and This is not a projection operator in the standard representation. In other words, and It's not the number of qubits, but the number of grid points ( Similarly, a corresponding state transition is required. Specifically, firstly, the state transition will be... Represented in the following binary form:

[0166] in, .

[0167] in this way, It can be represented as:

[0168] therefore,

[0169] Among them, symbols (primed product) is used for the first register (by...) Perform conventional matrix multiplication on the first register (composed of 100 qubits), while simultaneously performing matrix multiplication on the second register (composed of 100 qubits). The tensor product operation is performed on a quantum bit (composed of 100 qubits). In this embodiment, the above-mentioned... Named the window evolution operator, it is used to implement indexing of discrete grid points. Related, controlled evolution with windowed weights.

[0170] Figure 10 The window evolution operator is shown. The corresponding quantum circuit diagram.

[0171] On the other hand, targeting Introducing a quantum oracle. This quantum oracle is a unitary operation used in the field of quantum computing to encapsulate functions, specifically defined as follows:

[0172] The operation of the aforementioned quantum oracle is based on discrete grid index states. As input, the corresponding function value is output in the auxiliary register. While maintaining the input index state constant.

[0173] In addition, in order to ensure the above function values The accuracy required One auxiliary qubit, Encode it in binary form into an auxiliary register for subsequent implementation. The controlled power-law evolution provides directly readable quantum state forms. Regarding If required The higher the accuracy, the better. The larger the value (i.e., the more auxiliary qubits are needed).

[0174] Based on the quantum circuits of the above operators Figure 11 Furthermore, an approximate controlled evolution operator for a single time step is shown. A schematic diagram of a quantum circuit.

[0175] Then, in step ST500 (i.e., the fifth step), the approximate controlled evolution operator based on the above single time step is... The pre-defined quantum Fourier transform (QFT) unit and the aforementioned quantum oracle operator, i.e., the Oracle operator. The conjugate quantum oracle operator corresponding to this oracle is the conjugate oracle operator. And a pre-defined inverse quantum Fourier transform (IQFT) unit, to construct a quantum circuit that can run on a quantum computing device or a quantum simulator.

[0176] The quantum Fourier transform (QFT) unit is used to simulate the Fourier transform process, converting quantum states in the spatial domain to the frequency domain to simulate the spatial derivative terms of the discretized scalar conservation law equations, thereby reducing the computational complexity of the quantum circuit. On the other hand, the inverse quantum Fourier transform (IQFT) unit is used to inversely transform the quantum states in the frequency domain back to the spatial domain, restoring the spatial distribution characteristics of the scalar conservation law equations for subsequent quantum state measurements.

[0177] Furthermore, as explained above, quantum oracle operators (Oracle operators) Used to implement the coefficients corresponding to grid point indices in the difference scheme. The quantum encoding of the coefficients This is transformed into a quantum state form readable by quantum circuits, thus providing control signals for the evolution of the approximately controlled evolution operator. Conjugate quantum oracle operator. The above Oracle operators The conjugate transpose of is used to restore the auxiliary register to its initial state after the controlled evolution is completed, so as to counteract the state changes and phase coupling introduced by the operation of the Oracle operator, thereby maintaining the unitarity of the entire quantum circuit, avoiding distortion of quantum state evolution, and thus ensuring the accuracy of the solution results.

[0178] Figure 12A schematic diagram of a quantum circuit for solving scalar conservation law equations, constructed based on the aforementioned units or operators, is shown.

[0179] like Figure 12 As shown, the quantum circuit includes four quantum registers: index register R1, auxiliary register R2, deregister R3, and space register R4.

[0180] Index register R1 is used to store the index state of discrete grid points. (That is, its initial input quantum state is a grid-indexed state) This provides input control signals for Oracle operators. Furthermore, the index register R1 has... One quantum bit.

[0181] Auxiliary register R2 is initialized to all zeros. (That is, its initial input quantum state is an all-zero auxiliary state) ), used to store parameters output by Oracle operators. (parameter encoding state) This provides control signals for the approximate controlled evolution operator. Furthermore, after the controlled evolution is completed, it is restored to the initial all-zero state via the conjugate Oracle operator. Additionally, auxiliary register R2 has... One quantum bit.

[0182] Solution register R3 is used to store the initial solution state of the scalar conservation law equation. (That is, its initial input quantum state is the initial solution state) Furthermore, it stores the evolution results at each time step during the controlled evolution process. The solution register R3 has... One quantum bit.

[0183] Space register R4 is used to store auxiliary variables. quantum state corresponding to discrete spatial coordinates (That is, its initial input quantum state is the spatial ground state) The above quantum state The spatial domain and frequency domain are transformed sequentially through QFT and IQFT units, thereby simulating the spatial derivative term in the scalar conservation law equation. This space register R4 possesses... One quantum bit.

[0184] Oracle operators The bit lines connected simultaneously to index register R1 and auxiliary register R2 are in the initial state of index register R1. As input, initialize the auxiliary register R2. Convert to parameter state .

[0185] The QFT cell is connected to the bit line of space register R4, in space ground state. As input, it is converted from a spatial domain quantum state to a frequency domain quantum state.

[0186] Composite Approximate Controlled Evolution Operator Depend on The above-mentioned approximately controlled evolution operators are configured and connected to the bit lines of index register R1, auxiliary register R2, decomposition register R3, and space register R4, receiving the grid index state from index register R1. All-zero auxiliary state from auxiliary register R2 As a control signal, it receives the initial solution state from the solution register R3. and spatial ground state through QFT unit The transformed frequency-domain quantum state is used as input, and the solution quantum state at the target time T is output. (This can be simply referred to as the solution state at the target time) and the frequency-domain quantum state after controlled evolution (This can be simply referred to as the evolved frequency domain state) ).

[0187] The IQFT unit is connected to the output of the composite approximate controlled evolution operator. It receives the frequency domain quantum state after controlled evolution, performs an inverse transformation on it, and thus converts the evolved frequency domain state into the spatial domain quantum state after controlled evolution (which can be simply referred to as the target time spatial domain state). .

[0188] Then, in step ST600 (i.e., the sixth step), the initial quantum states of each register are set, and the process is executed. Figure 12 The quantum circuit shown is used to obtain the output quantum state of the quantum circuit.

[0189] First, the initial quantum states described above are prepared, and the corresponding initial quantum states are input into each of the aforementioned quantum registers. Specifically, the grid index state is input into the index register R1. Input all-zero auxiliary state into the auxiliary register Input the initial solution state into the solution register R3 And input the space ground state into space register R4. .

[0190] Secondly, parameter encoding and spatial transformation are performed. Specifically, the Oracle operator acts simultaneously on both the index register R1 and the auxiliary register R2, changing the grid index state. Corresponding coefficients The encoded data is stored in auxiliary register R2, and the output parameter encoding state is also displayed. On the other hand, the QFT unit acts on space register R4, setting the space ground state. It is converted into a frequency domain quantum state.

[0191] Then, the complete time evolution is performed. Specifically, the composite approximate controlled evolution operator acts simultaneously on the index register R1, auxiliary register R2, solution register R3, and space register R4, using the grid index state and parameter encoded state as control signals, the initial solution state and frequency domain quantum state as inputs, and outputting the solution quantum state at the target time and the evolved frequency domain quantum state, thereby completing the time evolution from the initial time to the target time.

[0192] Finally, the auxiliary register is reset and the state is reversed. Specifically, the conjugate Oracle operator acts on both the index register R1 and the auxiliary register R2, restoring the quantum state stored in the auxiliary register R2 to its initial all-zero auxiliary state. On the other hand, the IQFT unit acts on the space register R4, inversely transforming the evolved frequency-domain quantum state back into the spatial-domain quantum state, thus obtaining the spatial-domain quantum state at the target time.

[0193] By executing the above quantum circuit completely, the following output quantum state can be obtained:

[0194] in: It is the grid index state of index register R1; It is the all-zero auxiliary state after the auxiliary register R2 is reset; It is the solution quantum state at the target time of the solution register R3; It is the spatial domain state of the space register R4 at the target time after undergoing IQFT transformation; It is the discrete distribution function corresponding to the scalar conservation law equation. To solve space variables Grid index, For spatial location index.

[0195] Finally, in step ST700 (i.e., the seventh step), a measurement is performed based on the output quantum state to obtain a numerical solution to the scalar conservation law equation.

[0196] As a conventional measurement method, it is necessary to perform basis measurements on all qubits of the index register R1, the de-register R3, and the space register R4, and to calculate the probability of each ground state.

[0197] Then reconstruct the distribution function using probability:

[0198] Finally, the numerical solution of the scalar conservation law equation is obtained by weighted summation:

[0199] However, as shown in the expression above, the output quantum state is a multi-register superposition state, and the number of its ground states is:

[0200] As a result, an exponential number of measurements are required to obtain reasonably satisfactory accuracy, and the measurement complexity is:

[0201] in, This is a pre-set measurement precision. As can be seen, the measurement complexity increases exponentially with the number of qubits. Furthermore, after obtaining the measurement result, a relatively complex classical numerical integration is still required, further increasing the overall computational cost.

[0202] Therefore, this embodiment employs a measurement scheme combining quantum interference and controlled rotation, which enables the direct acquisition of numerical solutions without requiring the measurement of all registers. . Figure 13 The quantum circuit diagram corresponding to the measurement scheme used in this embodiment is shown. The main steps of this measurement scheme are as follows.

[0203] First, perform an Hadamard transform on index register R1. Specifically, apply an Hadamard gate to index register R1:

[0204] When the measurement result is In the (i.e., all-zero state) state, the deregister R3 collapses to:

[0205] Therefore, the summation result of the distribution function can be obtained directly without traversing all... .

[0206] Next, auxiliary bits are introduced, and controlled rotations are applied. Specifically, in order to obtain a weighted sum... Here, a single-bit auxiliary register is introduced (i.e., an auxiliary bit is introduced). A controlled rotation gate is applied to the k-th qubit of index register R1 and the aforementioned auxiliary bits. , where the rotation angle satisfy:

[0207] in, It is a sufficiently small constant to ensure that the small angle approximation holds true.

[0208] For any ground state After the controlled rotation operation on all bits, the state of the auxiliary bit becomes:

[0209] In particular, under small-angle approximation:

[0210] Therefore, the weighted sum above becomes:

[0211] Finally, the auxiliary qubit is measured to obtain the corresponding numerical solution. Specifically, the auxiliary qubit is measured to obtain its numerical value. The probability is:

[0212] Therefore, the following result can be obtained directly:

[0213] The above results represent the scalar conservation law equation at position. ,time Numerical solution:

[0214] According to the above measurement scheme, only the index register R1 and a single auxiliary qubit need to be measured, thus reducing the measurement complexity to:

[0215] Furthermore, after obtaining the above measurement results, the measurement scheme does not require classical post-processing and can directly output the measurement results as numerical solutions.

[0216] The computational method described in this embodiment, compared with traditional classical numerical methods, can overcome the curse of dimensionality in solving high-dimensional scalar conservation law equations. While ensuring numerical accuracy and physical conservation properties, it significantly reduces computational complexity and storage overhead, and improves solution efficiency and stability. Therefore, compared with classical numerical methods, the computational method described in this embodiment is applicable to higher-dimensional and larger-scale engineering physics scenarios.

[0217] (Description of a quantum computing system)

[0218] Based on the computation method described in the above embodiments, this disclosure also provides a quantum computing system according to an embodiment. Figure 14A schematic diagram of the main architecture of the quantum computing system described in this embodiment is shown.

[0219] The quantum computing system 900 architecture described in this embodiment is configured to perform numerical solutions to the above-mentioned scalar conservation law equations, such as... Figure 14 As shown, the quantum computing system 900 mainly includes a first module 901, a second module 902, a third module 903, a fourth module 904, a fifth module 905, a sixth module 906, and a seventh module 907.

[0220] The first module 901 is configured to convert the scalar conservation law equation into a linear Liouville equation through a level set dimension-up transformation.

[0221] The second module 902 is configured to spatially discretize the linear Liouville equation using the finite difference method, transforming it into a non-Hermitian matrix system.

[0222] The third module 903 is configured to perform Schrödingerization on the pairwise non-Hermitian matrix system to construct a Hermitian Hamiltonian system with Hermitian Hamiltonian quantities.

[0223] The fourth module 904 is configured to construct a continuous controlled evolution operator based on the Hermi-Hamiltonian, and to transform the continuous controlled evolution operator into an approximate controlled evolution operator through Li-Trott-Suzuki decomposition.

[0224] The fifth module 905 is configured to construct a quantum circuit that can run on a quantum processor or quantum simulator based on the approximate controlled evolution operator, the preset QFT unit, the Oracle operator, the conjugate Oracle operator, and the IQFT unit.

[0225] The sixth module 906 is configured to prepare the initial quantum state of the quantum circuit and execute the quantum circuit based on the initial quantum state to obtain the output quantum state of the quantum circuit.

[0226] The seventh module 907 is configured to measure the output quantum state and determine the numerical solution of the scalar conservation law equation based on the measurement results.

[0227] It should be understood that the above modules can be implemented in the form of software, firmware, hardware, or any combination thereof. For example, as a specific implementation, the first, second, third, fourth, fifth, and seventh modules can be software / firmware modules running on a classical computer and executed by a classical processor, while the sixth module can be an interface module controlling the quantum execution end, issuing instructions from the classical processor to the quantum processor or quantum simulator to realize quantum state preparation, quantum circuit execution, and measurement of the output quantum state.

[0228] On the other hand, it should also be understood that the quantum computing system 900 described in this embodiment can be implemented in various product forms. For example, it can be a software toolkit or application running on a classical computer, a quantum computing service platform deployed on a cloud server, control software integrated into a quantum computing device, or a numerical calculation and circuit compilation module in a quantum simulator.

[0229] It should also be noted that the above module division is only a logical functional division. In actual implementation, the specific division of modules can be adjusted according to the actual situation.

[0230] (Description of computing device 1000)

[0231] Based on the calculation method described in the above embodiments, this disclosure also provides a calculation device according to an embodiment. Figure 15 A schematic diagram of the hardware architecture of the computing device described in this embodiment is shown.

[0232] The computing device 1000 described in this embodiment is configured to perform numerical solutions to the aforementioned scalar conservation law equations. For example... Figure 15 As shown, the computing device 1000 mainly includes a quantum processor 1001, a classical processor 1002, and a memory 1003. The memory 1003 is communicatively connected to the classical processor 1002, and the classical processor 1002 is communicatively connected to the quantum processor 1001, thereby enabling the issuance of control commands and the transmission of measurement results.

[0233] The memory 1003 is, for example, a non-volatile storage medium and stores computer-executable instructions (or computer programs). When these computer-executable instructions are executed by the classical processor 1002, the computing device 1000 can implement the computational methods described in the above embodiments. Specifically, the executable instructions stored in the memory 1003 include instructions for level set dimensionality upgrading, finite difference discretization, Schrödingerization, Hamiltonian construction and decomposition, quantum circuit construction, quantum circuit execution control, and measurement result post-processing instructions, etc.

[0234] The classic processor 1002 is the main control unit of the computing device 1000 and is configured to perform the following operations: Read and execute the computer-executable instructions from the memory 1003; The scalar conservation law equation is transformed into a linear Liouville equation by using the level set dimension-up transformation; The linear Liouville equation is spatially discretized using the finite difference method to obtain a non-Hermitian matrix system; Schrödingerization is applied to the non-Hermitian matrix system to construct the Hermitian Hamiltonian system; A continuous controlled evolution operator is constructed based on the Hermi-Hamiltonian, and an approximate controlled evolution operator is obtained through Li-Trott-Suzuki decomposition. Based on the approximate controlled evolution operator, the pre-defined QFT unit, the Oracle operator, the conjugate Oracle operator, and the IQFT unit, a quantum circuit that can run on the quantum processor 1001 or a quantum simulator is constructed. Control commands are sent to the quantum processor 1001 to control it to prepare an initial quantum state, execute quantum circuits, and complete quantum measurements. The measurement results are obtained from the quantum processor 1001, and statistical analysis and post-processing are performed to determine the numerical solution of the scalar conservation law equation.

[0235] The classic processor 1002 can be a general-purpose central processing unit (CPU), graphics processing unit (GPU), field-programmable gate array (FPGA), or application-specific integrated circuit (ASIC), and is mainly responsible for algorithm logic processing, quantum circuit compilation, quantum processor control, and post-processing of measurement results.

[0236] Quantum processor 1001 is the quantum execution unit of computing device 1000, configured to perform the following quantum operations under the control of classical processor 1002: The initial quantum state of the quantum circuit is prepared according to the instructions issued by the classical processor 1002; The quantum evolution operation is performed according to the gate sequence of the quantum circuit to obtain the output quantum state; Perform physical measurements on the output quantum state to obtain measurement results in the form of a bit string; The measurement results are then sent back to the classical processor 1002.

[0237] The quantum processor 1001 can be a superconducting quantum chip, an ion trap quantum processor, a photonic quantum computing device, or a quantum simulator. It is mainly responsible for the preparation, evolution, and measurement of quantum states, and does not participate in classical computational processes such as equation transformation, discretization, and compilation.

[0238] In this embodiment, the workflow of the computing device 1000 is as follows: The computer program in the memory 1003 is read and executed by the classical processor 1002; The classical processor 1002 performs equation transformation, discretization, Schrödingerization, Hamiltonian construction and decomposition, and quantum circuit construction. The classical processor 1002 sends control commands to the quantum processor 1001 to control it to prepare an initial quantum state and execute quantum circuits; The quantum processor 1001 performs quantum evolution and measurement, and sends the measurement results back to the classical processor 1002; The classical processor 1002 performs post-processing based on the measurement results to obtain a numerical solution to the scalar conservation law equation.

[0239] The computing device 1000 described in this embodiment can be implemented in various product forms. For example, the computing device 1000 can be a dedicated quantum computing device integrating a classical control unit (i.e., a classical processor) and a quantum processing unit (i.e., a quantum processor), a hybrid computing system composed of a classical server and a quantum chip, a quantum computing service node deployed in the cloud, or a high-performance computing device equipped with a quantum simulator.

[0240] It should be noted that the above architecture is only one specific implementation of this disclosure and is not limited thereto. The classical processor 1002, the quantum processor 1001, and the memory 1003 can be physically independent devices or integrated into the same physical device.

[0241] (Numerical experimental verification)

[0242] To further verify the technical feasibility, solution accuracy, numerical convergence, and hardware feasibility of the scalar conservation law equation calculation method described in this embodiment, numerical experiments were conducted based on the quantum simulation platform UnitaryLab (registered trademark). The one-dimensional inviscid Burgers equation in the field of fluid mechanics and the one-dimensional LWR traffic flow equation in the field of traffic engineering were solved and verified respectively. Periodic boundary conditions were used in the experiments, and the experimental parameters and symbol definitions were consistent with those in the above embodiment.

[0243] Specifically, in this numerical experiment, the computational domain is considered. The above questions:

[0244] On the one hand, consider And adopt Gaussian initial conditions (i.e. The equation to be solved is: This is the inviscid Burgers equation. On the other hand, consider... And take initial conditions Therefore, the equation to be solved is a classic traffic flow model.

[0245] For the inviscous Burgers equation, the normalized target total duration is taken. , , , The calculations were performed using the calculation methods described in the above embodiments (the quantum simulation method (LTS decomposition + Schrödingerization) described in the above embodiments), the classical Schrödingerization method, and the finite difference matrix multiplication method, respectively. Figure 16 Different A graph showing the numerical solution of the inviscid Burgers equation, where the horizontal axis represents the spatial location of the equation within the computational domain, and the vertical axis represents the numerical solution at each spatial location within the computational domain at the target total duration T. .Should Figure 16 The following diagrams illustrate different results obtained using the computational methods described in this disclosure (also known as quantum simulation methods based on quantum gates, yellow, labeled Trotter Schro), the classical Schrödinger method (also known as the direct integral Schrödinger method, blue, labeled Classical Schro), and the finite difference matrix multiplication method (red, labeled Matrix operator). The fitting curve of the numerical solution of the inviscid Burgers equation is obtained. On the other hand, for the classical traffic flow model, the normalized target total duration is taken. , , , The calculation is performed using the calculation method described in the above embodiments. Figure 17 The following diagrams illustrate different results obtained using the calculation method described in this disclosure (yellow), the classical Schrödinger method (blue), and the finite difference matrix product method (red). The curve of the numerical solution of the classical traffic flow model.

[0246] observe Figure 16 and 17 It can be seen that, with the increase of the number of auxiliary qubits... With the increase of the number of auxiliary qubits, the accuracy of the numerical solution obtained by the calculation method described in this disclosure is significantly improved. More specifically, when the number of auxiliary qubits is small... At that time, the fitting curve formed by the numerical solution obtained by the calculation method described in this disclosure exhibits obvious oscillations and non-physical artifacts (this phenomenon is particularly prominent in traffic flow models), and its solution fails to effectively capture a smooth sine waveform in the initial stage. However, with the increase of auxiliary qubits... When the value is gradually increased to 6, the numerical solution obtained by the calculation method described in this disclosure and the numerical solution obtained by the finite difference matrix product method are basically completely coincident.

[0247] Furthermore, by comparing the numerical solutions obtained by the classical Schrödingerization method (based on direct integration) with the fitting curves of the computational method described in this disclosure (a quantum simulation method based on quantum gates, including LTS decomposition and Schrödingerization processing), it can be seen that with the number of auxiliary qubits... In this case, the error of the numerical solution obtained based on the calculation method described in this disclosure is well controlled.

[0248] on the other hand, Figure 18A schematic diagram is shown illustrating the analysis results of quantum gate complexity when solving the above-mentioned inviscid Burgers equation using the computational method described in this disclosure. (a) shows the relationship curve between the number of auxiliary qubits and the total number of quantum gates on a linear scale, and (b) shows the relationship curve between the number of auxiliary qubits and the total number of quantum gates on a logarithmic scale. According to the above analysis results, the total number of quantum gates in the computational method described in this disclosure increases with the number of spatial qubits. and auxiliary qubit number It grows exponentially. That is, according to the calculation method described in this disclosure, the curse of dimensionality caused by the exponential growth of the number of discrete grid points with the increase of dimension in existing calculation methods in high-dimensional scenarios can be avoided.

[0249] also, Figure 19 The second-order norm error of the number of auxiliary qubits and the accuracy of the numerical solution is shown when solving the above-mentioned inviscid Burgers equation using the computational method described in this disclosure. The relationship between them. In particular, before the calculation, the number of spatial qubits is... Set the time step to 5. (Right now Set the truncation value to 0.005 and the cutoff length R to 4. It is known that when introducing the Schrödinger method, the variables... Apply a distorted phase .for , The natural extension form is It can be seen that the function is in It is not differentiable, that is .according to Figure 19 The results show that the error convergence rate at this point is only approximately... On the other hand, if we change to using Defined in Then the convergence rate can be improved to approximately ,like Figure 19 As shown. Theoretically speaking, if If the surface is sufficiently smooth, the error can achieve spectral convergence.

[0250] The above description is merely a preferred embodiment of this disclosure and is not intended to limit the scope of protection of this disclosure. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this disclosure should be included within the scope of protection of this disclosure.

[0251] After understanding the technical concept and implementation methods of this disclosure, those skilled in the art can make appropriate changes and modifications without departing from the spirit of this disclosure. Such changes and modifications should also be considered as equivalent implementation methods of this disclosure and fall within the protection scope of the appended claims.

Claims

1. A method for calculating scalar conservation law equations, characterized in that, include: The first step is to transform the scalar conservation law equation into a linear Liouville equation through a level set dimension-up transformation. The second step is to spatially discretize the linear Liouville equation using the finite difference method, transforming it into a non-Hermitian matrix system. The third step is to perform Schrödingerization on the non-Hermitian matrix system to construct a Hermitian Hamiltonian system with Hermitian Hamiltonian. The fourth step is to construct a continuous controlled evolution operator based on the Hermi-Hamiltonian, and then transform the continuous controlled evolution operator into an approximate controlled evolution operator through Li-Trott-Suzuki decomposition. The fifth step involves constructing a quantum circuit that can run on a quantum processor or quantum simulator based on the approximate controlled evolution operator, the preset QFT unit, the Oracle operator, the conjugate Oracle operator, and the IQFT unit. The sixth step is to prepare the initial quantum state of the quantum circuit and execute the quantum circuit based on the initial quantum state to obtain the output quantum state of the quantum circuit. as well as The seventh step is to measure the output quantum state and determine the numerical solution of the scalar conservation law equation based on the measurement results.

2. The calculation method as described in claim 1, characterized in that, The first step includes: Regarding the scalar conservation law equation ， Define the level set function ， in, It is a scalar conserved field to be solved. It is a spatial variable. It is a time variable. It is the flux function. These are solution space variables introduced to perform the aforementioned level set dimensionality-up transformation; Substituting the level set function into the scalar conservation law equation transforms the scalar conservation law equation into a first-order linear transport equation. ;as well as The level set function is transformed into the corresponding distribution function, thereby transforming the first-order linear transport equation into the linear Liouville equation. 。 3. The calculation method as described in claim 2, characterized in that, The second step includes: For the linear Liouville equation, the spatial variables and the solution space variables are discretized into a grid using the finite difference method to form a time evolution distribution function vector; and Based on the discretized spatial variables and solution spatial variables, the linear Liouville equation is transformed into a time-continuous non-Hermitian matrix system: ， in, It is the time evolution distribution function vector, It is a non-Hermitian matrix.

4. The calculation method as described in claim 3, characterized in that, The third step includes: For the time evolution distribution function vector Introducing auxiliary variables To construct augmented variables ; For the non-Hermitian matrix Decompose to form the first Hermitian matrix. Second Hermitian matrix ,satisfy ， in, It is the non-Hermitian matrix The conjugate transpose of ; Based on the augmented variables and the first Hermitian matrix and the second Hermitian matrix Constructing an intermediate evolution matrix system ;as well as Perform a Fourier transform on the auxiliary variables in the augmented variables to convert the intermediate evolution matrix system into the Hermi-Hamiltonian system. ， in, It is the auxiliary variable Fourier transform conjugate variable, It is the Hermi-Hamiltonian.

5. The calculation method as described in claim 4, characterized in that, The fourth step includes: Construct a continuously controlled evolution operator per unit time step based on the Hermi-Hamiltonian. ,satisfy ， in, It refers to the unit time step; The Hermi-Hamiltonian Decomposed into central difference Hamiltonian components and symmetric Hamiltonian components ; Based on the central difference Hamiltonian components Construct the following first basic controlled evolution operator ； Based on the symmetric Hamiltonian components Construct the following second fundamental controlled evolution operator ;as well as The continuous controlled evolution operator is approximated as a product of the first and second basic controlled evolution operators through the Li-Trott-Suzuki decomposition, thus obtaining the approximate controlled evolution operator. The first and second basic controlled evolution operators can be realized by a combination of quantum gates, such as the Adama gate, phase gate, controlled NOT gate, and Pauli X gate.

6. The calculation method as described in claim 5, characterized in that, The first basic controlled evolution operator include: The first global phase correction term is used to apply a specific phase shift operation to all qubits corresponding to the spatial variable; The first many-body global flip evolution term is used to realize the many-body interaction of simultaneously flipping all qubits corresponding to the spatial variable; and The first individual unit evolution term is used to realize the independent evolution of each qubit corresponding to the spatial variable. The second basic controlled evolution operator include: The second many-body global flip evolution term is used to realize the many-body interaction with antisymmetric phase characteristics of all qubits corresponding to the spatial variable; and The second single-unit rise-fall evolution term is used to realize the independent antisymmetric rise-fall evolution of each qubit corresponding to the spatial variable.

7. The calculation method as described in claim 5, characterized in that, In the quantum circuit, A composite controlled evolution operator is constructed based on the evolution duration, the unit time step, and the approximately controlled evolution operator of the unit time step. The initial quantum state includes the grid index state corresponding to the solution space variables, the preset all-zero auxiliary state, the initial solution state corresponding to the initial conditions of the scalar conserved field to be solved, and the space ground state corresponding to the auxiliary variables. In the sixth step, The quantum gates corresponding to the Oracle operator in the quantum circuit are combined to receive and process the grid index state and the all-zero auxiliary state, transforming the all-zero auxiliary state into a parameter-encoded state and outputting it. The quantum gate combination corresponding to the QFT unit in the quantum circuit receives and processes the spatial ground state, and outputs the corresponding frequency domain quantum state. The quantum gate combination corresponding to the composite controlled evolution operator in the quantum circuit receives and processes the grid index state, the parameter-encoded state, the initial solution state, and the frequency domain quantum state, and outputs the solution quantum state and the evolved frequency domain quantum state at the target time. The quantum gates in the quantum circuit corresponding to the conjugate Oracle operator are combined to receive the grid index state and the parameter-encoded state, and the parameter-encoded state is transformed back into the all-zero auxiliary state. The quantum gate combination corresponding to the IQFT unit in the quantum circuit receives and processes the evolved frequency domain quantum state, so as to transform the evolved frequency domain quantum state into the spatial domain quantum state at the corresponding target time.

8. The calculation method as described in claim 7, characterized in that, The output quantum state is a combination of the grid index state, the all-zero auxiliary state, the solution quantum state at the target time, the spatial domain quantum state at the target time, and the discrete distribution function formed by discretizing the distribution function: in, It is the grid index state, It is the all-zero auxiliary state. It is the solution quantum state at the target time. It is the spatial domain quantum state at the target time. It is the discrete distribution function, For the grid index of the solution space variables, The spatial location index is the spatial variable, and the discrete distribution function represents time. Spatial variables Spatial location, solution space variables Grid index The function value at that point, In the seventh step: An Hadamard transform is performed on the grid index state so that, in the case that the measurement result obtained after measuring the grid index state is a completely zero state, the solution quantum state at the target time collapses into the sum of the discrete distribution function: in, It is the number of discrete grids for the spatial variable. It is the number of discrete grids of the solution space variables. It indicates the time. Spatial location Solution space variables Grid index Discrete function values ​​at; An auxiliary bit is introduced, and a controlled rotation transformation is performed on the auxiliary bit and each grid index state to convert the summation result into a combination of the following quantum states. in, It is a preset constant that satisfies ;as well as The auxiliary bit is measured, and based on the measurement, the auxiliary bit is obtained as follows: The probability is used to determine the numerical solution of the scalar conservation law equation.

9. A quantum computing system capable of solving scalar conservation law equations, characterized in that, include: The first module is configured to convert the scalar conservation law equation into a linear Liouville equation through a level set dimension-up transformation. The second module is configured to spatially discretize the linear Liouville equation using the finite difference method, transforming it into a non-Hermitian matrix system. The third module is configured to perform Schrödingerization on non-Hermitian matrix systems to construct Hermitian Hamiltonian systems with Hermitian Hamiltonian quantities. The fourth module is configured to construct a continuous controlled evolution operator based on the Hermi-Hamiltonian, and to transform the continuous controlled evolution operator into an approximate controlled evolution operator through Li-Trott-Suzuki decomposition; The fifth module is configured to construct quantum circuits that can run on a quantum processor or quantum simulator based on the approximate controlled evolution operator, preset QFT units, Oracle operators, conjugate Oracle operators, and IQFT units. The sixth module is configured to prepare the initial quantum state of the quantum circuit and execute the quantum circuit based on the initial quantum state to obtain the output quantum state of the quantum circuit; as well as The seventh module is configured to measure the output quantum state and determine the numerical solution of the scalar conservation law equation based on the measurement results.

10. A computing device for solving scalar conservation law equations, comprising a quantum processor, a classical processor, and a memory, wherein the memory stores a computer program, characterized in that, When the computer program is executed by the classical processor, it performs the following steps: By using the level set dimension-up transformation, the scalar conservation law equation is transformed into a linear Liouville equation; The linear Liouville equation is spatially discretized using the finite difference method, transforming it into a non-Hermitian matrix system. Schrödingerization is performed on the non-Hermitian matrix system to construct a Hermitian Hamiltonian system with Hermitian Hamiltonian; A continuous controlled evolution operator is constructed based on the Hermi-Hamiltonian, and the continuous controlled evolution operator is transformed into an approximate controlled evolution operator through Li-Trott-Suzuki decomposition. Based on the approximate controlled evolution operator, the preset QFT unit, the Oracle operator, the conjugate Oracle operator, and the IQFT unit, a quantum circuit that can run on a quantum computing device or a quantum simulator is constructed. The quantum processor is controlled to prepare the initial quantum state of the quantum circuit, and the quantum circuit is loaded and executed to obtain the output quantum state of the quantum circuit; as well as The quantum processor is controlled to perform a measurement on the output quantum state, the measurement result is extracted from the quantum processor, and the numerical solution of the scalar conservation law equation is determined based on the measurement result.

11. A computer-readable storage medium storing a computer program, characterized in that, The computer program is executed by a processor to implement the method for calculating the scalar conservation law equation as described in any one of claims 1 to 8.

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