5/10/2023 0 Comments Solving complex 4 equation systems![]() 2a, we study the scaling behavior for κ = 1.1 and R = 2. We conclude by applying the algorithm iteratively on a fixed problem and study the convergence of the relative residual. In the following section, we demonstrate the scaling of the annealing algorithm under varying problem size, condition number, and precision of the search space. ![]() A common right-hand side is taken for all P (1): a vector of length N with linearly spaced decimals between 1 and −1. We end by using quantum annealing to solve a similar system and characterize the performance from experimental demonstrations with a commercial quantum annealer. However, well-known numerical challenges slow convergence with conventional solvers 6, 7. The solution to the discretized Dirac equation is currently the only approach for evaluating non-perturbative QCD. We then narrow the scope to examples of linear equations by first demonstrating an application to linear regression, before elucidating results on ill-conditioned linear systems motivated by the discretized Dirac equation Dϕ = χ from lattice quantum chromodynamics (QCD). In this work, we present an approach for solving a general system of n th-order polynomial equations based on the principles of quantum annealing, followed by a demonstration of the algorithm for a system of second-order polynomial equations on commercially available quantum annealers. Additionally, an algorithm for solving linear systems within the adiabatic quantum computing model 3 was experimentally demonstrated 4, followed by a more recent proposal 5. For example, a quantum algorithm for solving systems of linear equations was established for gate-based quantum computers 1 and demonstrated with small-scale problem instances 2. The advent of quantum computing has opened up the possibility of new methods for solving these challenging problems. ![]() Conventional methods for solving linear systems range from exact methods, such as matrix diagonalization, to iterative methods, such as fixed-point solvers, while polynomial systems are typically solved iteratively with homotopy continuation. Many problems in science, engineering, and mathematics can be reduced to solving systems of equations with notable examples in modeling and simulation of physical systems, and the verification and validation of engineering designs. ![]() Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10 −8. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. Numerous scientific and engineering applications require numerically solving systems of equations.
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