Linear Matrix Equation Solver Using Unitary Decomposition
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Solution Overview
Problem
Current methods for solving linear matrix equations, particularly in computational fluid dynamics (CFD) and finite element analysis (FEA), are inefficient due to the complexity of decomposing large matrices into unitary matrices, limiting the use of quantum computing devices to smaller, simpler matrices.
Innovation Solution
The method involves determining a linear combination of unitary matrices equivalent to the matrix A, updating coefficients based on the solution vector, and using this combination to solve updated linear matrix equations, exploiting the fixed sparsity pattern of matrices in CFD and FEA problems to reduce computational expense.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If quantum computing devices are used to solve linear matrix equations, then computational efficiency is improved, but the complexity of decomposing large matrices into unitary matrices increases
Solution Approach 1:
The patent segments the large matrix A into multiple smaller unitary matrices through decomposition. By breaking down the complex matrix into manageable unitary components, the system can process each segment independently on quantum computing devices, reducing the overall computational complexity while maintaining the ability to solve large-scale linear matrix equations efficiently
Solution Approach 2:
The patent performs preliminary decomposition of the matrix A into unitary matrices before the actual quantum computing process. This pre-processing step prepares the matrix in a form suitable for quantum algorithm execution, separating the complex decomposition task from the iterative solving process and enabling more efficient subsequent computations
2Measurement precision
If iterative methods are used to solve non-linear problems, then solution accuracy is improved, but the number of iterations and computational time increase
Solution Approach 1:
The patent performs preliminary decomposition of the matrix into unitary components before the iterative process begins. This pre-computation remains valid across multiple iterations, allowing the system to quickly update solutions without repeating the full decomposition, thereby reducing computational time while maintaining solution accuracy through iterative refinement
Solution Approach 2:
The patent dynamically updates only the necessary components of the matrix decomposition in each iteration rather than recomputing everything from scratch. By adapting the pre-decomposed unitary matrices to the current iteration's requirements, the system achieves both fast computation and accurate solutions
Data Source
AI summary
A method, performed on at least one computing device, of determining solutions to a number of linear matrix equations satisfying A{right arrow over (x)}={right arrow over (b)}, where A is a n×n matrix, {right arrow over (x)} is a column vector with n entries, and {right arrow over (b)} is a column vector with n entries, is disclosed. The method comprises determining a linear combination of unitary matrices that is equivalent to the matrix A; based on the linear combination of unitary matrices, determining a column vector {right arrow over (x)} that satisfies the linear matrix equation; forming an updated matrix A based on the obtained column vector {right arrow over (x)}; forming an updated column vector {right arrow over (b)} based on the obtained column vector {right arrow over (x)}; updating the coefficients of the linear combination of unitary matrices based on the updated column vector {right arrow over (x)}; and based on the updated linear combination of unitary matrices, determining an updated column vector {right arrow over (x)} that satisfies the updated linear matrix equation.