KR-20260068017-A - METHOD FOR SOLVING DIFFERENTIAL EQUATION USING QUANTUM COMPUTING AND APPARATUS THEREOF

KR20260068017AKR 20260068017 AKR20260068017 AKR 20260068017AKR-20260068017-A

Abstract

The present invention includes a method for calculating a differential equation comprising: a step of converting a partial differential equation (PDE) into an ordinary differential equation (ODE) through spectral expansion; a step of estimating the solution of the converted ordinary differential equation (ODE) using a quantum circuit; a step of constructing a Hamiltonian by applying a time discretization technique to the ordinary differential equation; a step of calculating eigenvalues and eigenvectors of the ordinary differential equation using the Hamiltonian and the measurement results of the quantum circuit; and a step of restoring the solution of the partial differential equation using the eigenvalues and eigenvectors.

Inventors

강두형
이스마에일리파 이스마에일
유필선
김경민
이증락

Assignees

주식회사 큐노바

Dates

Publication Date: 20260513
Application Date: 20251106
Priority Date: 20241106

Claims (20)

In a method for calculating differential equations performed by a computing device, A step of transforming a Partial Differential Equation (PDE) into an Ordinary Differential Equation (ODE) through spectral expansion; A step of estimating the solution of the transformed ordinary differential equation (ODE) using a quantum circuit; A step of constructing a Hamiltonian by applying a time discretization technique to the above ordinary differential equation; A step of calculating eigenvalues and eigenvectors of the ordinary differential equation using the above Hamiltonian and the measurement results of the above quantum circuit; and A method for calculating a differential equation comprising the step of restoring the solution of the partial differential equation using the eigenvalues and eigenvectors.
In paragraph 1, the conversion step is, A method for calculating a differential equation characterized by using a Fourier series or a Chebyshev polynomial as a basis function for expanding a spatial variable in the above-mentioned spectral expansion.
In paragraph 2, A method for calculating a differential equation that further includes the step of inputting spatial information and coefficient information of the above basis function into the above quantum circuit.
In paragraph 1, A method for calculating differential equations characterized by using Backward Euler or Crank-Nicolson as the time discretization technique.
In claim 1, the Hamiltonian configuration step is, A step of calculating a linear equation by applying a time discretization technique to the above ordinary differential equation; and A method for calculating a differential equation characterized by including the step of constructing a Hamiltonian based on the linear equation calculated above.
In paragraph 5, the linear equation calculation step is, A method for calculating a differential equation characterized by utilizing coefficient information of a basis function calculated in a previous time step to calculate a linear equation.
In claim 1, the step of calculating the eigenvalues and eigenvectors is, A step of constructing a subspace Hamiltonian using the above Hamiltonian and the measurement results of the above quantum circuit; and A method for calculating a differential equation characterized by including the step of calculating the eigenvalues and eigenvectors of the ordinary differential equation by diagonalizing the above subspace Hamiltonian.
In claim 7, the above subspace Hamiltonian configuration step is, A method for calculating a differential equation characterized by selecting bitstrings with high measurement probabilities among a plurality of bitstrings obtained from the above quantum circuit to create a subspace, and constructing a subspace Hamiltonian by projecting a Hamiltonian onto the said subspace.
In Paragraph 7, A step of calculating a cost function (C) based on the residual (r) of the linear equation calculated through the above time discretization technique; and A method for calculating a differential equation that further includes a step of determining whether the above cost function converges.
In Paragraph 9, A method for calculating a differential equation that further includes the step of optimizing the parameters (θ) of the quantum circuit to minimize the cost function when the cost function does not converge, and re-driving the quantum circuit based on the optimized parameters (θ).
In Paragraph 9, When the above cost function converges, a step of calculating a truncation error using coefficient information estimated in the above quantum circuit; and A method for calculating a differential equation that further includes the step of checking whether the calculated cutting error is less than or equal to a preset threshold.
In Paragraph 11, A method for calculating a differential equation that further includes the step of reconstructing the subspace Hamiltonian by expanding the dimension of the subspace when the above truncation error is greater than the above threshold.
A computer program stored on a computer-readable recording medium so that a method according to any one of claims 1 to 12 can be executed on a computer.
In a differential equation computing device comprising one or more processors, The above one or more processors are: The operation of transforming a Partial Differential Equation (PDE) into an Ordinary Differential Equation (ODE) through spectral expansion; An operation to estimate the solution of the transformed ordinary differential equation (ODE) using a quantum circuit; The operation of constructing a Hamiltonian by applying a time discretization technique to the above ordinary differential equation; The operation of calculating the eigenvalues and eigenvectors of the ordinary differential equation using the above Hamiltonian and the measurement results of the above quantum circuit; and A differential equation calculator that performs the operation of restoring the solution of the partial differential equation using the above eigenvalues and eigenvectors.
In Clause 14, the above conversion operation is, A differential equation calculator characterized by using a Fourier series or a Chebyshev polynomial as a basis function for expanding a spatial variable in the above-mentioned spectral expansion.
In Clause 14, the above Hamiltonian configuration operation is, The operation of deriving a linear equation by applying a time discretization technique to the above ordinary differential equation; and A differential equation calculator characterized by including an operation to construct a Hamiltonian based on the linear equation derived above.
In Clause 14, the operation of calculating the eigenvalues and eigenvectors above is, An operation to construct a subspace Hamiltonian using the above Hamiltonian and the measurement results of the above quantum circuit; and A differential equation calculator characterized by including the operation of calculating eigenvalues and eigenvectors of the ordinary differential equation by diagonalizing the above subspace Hamiltonian.
In paragraph 17, the above subspace Hamiltonian configuration operation is, A differential equation computing device characterized by generating a subspace by selecting bitstrings with high measurement probabilities among a plurality of bitstrings obtained from the above quantum circuit, and constructing a subspace Hamiltonian by projecting a Hamiltonian onto the said subspace.
In paragraph 14, the above one or more processors, The operation of calculating a cost function (C) based on the residual (r) of the linear equation calculated through the above time discretization technique; and A differential equation calculator that further performs the operation of determining whether the above cost function converges.
In paragraph 19, the above one or more processors, A differential equation computing device that, when the above cost function does not converge, further performs the operation of optimizing the parameters (θ) of the quantum circuit to minimize the above cost function and re-driving the quantum circuit based on the optimized parameters (θ).

Description

Method for Solving Differential Equations Using Quantum Computing and Apparatus Thereof The present invention relates to quantum computing technology, and more specifically, to a method and apparatus for solving partial differential equations (PDEs) using quantum computing. Quantum computers are a new concept of computer capable of processing multiple pieces of information simultaneously by utilizing the unique physical properties of quantum mechanics, such as superposition and entanglement. The need for quantum computers is gradually increasing as an alternative to overcome the performance limitations of classical computers caused by leakage currents occurring in the microcircuits of modern semiconductor chips. Through quantum parallel processing, which uses quantum bits or qubits—quantum units of information—as the basic unit of information processing, quantum computers enable information processing and computation speeds to increase exponentially, allowing them to solve problems at high speeds. Accordingly, quantum computers are expected to bring massive innovation to various industrial sectors, including finance, chemistry, and pharmaceuticals, as they excel in complex calculations and large-scale data processing, such as optimal pathfinding, prime factorization, and large-scale data exploration. Meanwhile, Partial Differential Equations (PDEs) are widely used to describe the behavior of physical systems that change over time and space. To find the solutions to these PDEs, it is necessary to numerically approximate the differential terms and convert them into a system of linear equations for calculation. Conventional numerical analysis techniques perform these calculations on classical computers and primarily utilize linear solution methods based on matrix operations (e.g., Lower-Upper decomposition, QR decomposition, iterative methods, etc.). However, there are limitations in that computational complexity and memory requirements increase exponentially when high-dimensional PDE problems or non-linear terms are involved, and computational efficiency significantly degrades in problems with long time progressions. Therefore, a method to compute solutions to partial differential equations using quantum computers is required. FIG. 1 is a diagram showing the configuration of a differential equation calculation device according to one embodiment of the present invention; FIG. 2 is a diagram showing the configuration of the classical computing module of FIG. 1; FIG. 3 is a diagram showing the configuration of the quantum computing module of FIG. 1; FIG. 4 is a flowchart illustrating a method for calculating differential equations according to an embodiment of the present invention; FIG. 5 is a diagram referenced to explain the method of calculating the differential equation of FIG. 4; FIG. 6 is a block diagram of a computing device according to an embodiment of the present invention. Hereinafter, embodiments disclosed in this specification will be described in detail with reference to the attached drawings. Identical or similar components regardless of drawing symbols will be assigned the same reference number, and redundant descriptions thereof will be omitted. The suffixes "module" and "part" for components used in the following description are assigned or used interchangeably solely for the ease of drafting the specification and do not inherently possess distinct meanings or roles. That is, the term "part" used in this invention refers to a hardware component such as software, FPGA, or ASIC, and the "part" performs certain roles. However, the meaning of "part" is not limited to software or hardware. The "part" may be configured to reside in an addressable storage medium or may be configured to run one or more processors. Accordingly, as an example, a 'part' includes components such as software components, object-oriented software components, class components, and task components, as well as processes, functions, attributes, procedures, subroutines, segments of program code, drivers, firmware, microcode, circuits, data, databases, data structures, tables, arrays, and variables. The functionality provided within the components and 'parts' may be combined into a smaller number of components and 'parts' or further separated into additional components and 'parts'. In addition, when describing the embodiments disclosed in this specification, if it is determined that a detailed description of related prior art may obscure the essence of the embodiments disclosed in this specification, such detailed description is omitted. Furthermore, the attached drawings are intended only to facilitate understanding of the embodiments disclosed in this specification, and the technical concept disclosed in this specification is not limited by the attached drawings; it should be understood that the drawings include all modifications, equivalents, and substitutions that fall within the spirit and technical scope of the invention. Th