Annealing-based approach to solving partial differential equations
This work addresses the challenge of solving PDEs on specialized hardware (e.g., quantum annealers) by reformulating the problem as an eigenvalue optimization, but the approach is incremental as it relies on iterative simulated annealing and lacks concrete performance comparisons.
The paper proposes an annealing-based method for solving partial differential equations by converting the discretized linear system into a generalized eigenvalue problem, which is then solved via optimization of a Rayleigh quotient. Simulated annealing results show that the number of iterations scales with system size and annealing time, enabling arbitrary precision without increasing variables.
Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general eigenvalue problem, which can be transformed into an optimization problem with an objective function given by a generalized Rayleigh quotient. The proposed algorithm requires iterative computations. However, it enables efficient annealing-based computation of eigenvectors to arbitrary precision without increasing the number of variables. Investigations using simulated annealing demonstrate how the number of iterations scales with system size and annealing time. Computational performance depends on system size, annealing time, and problem characteristics.