Variational Quantum Framework for Nonlinear PDE Constrained Optimization Using Carleman Linearization

We present a novel variational quantum framework for nonlinear partial differential equation (PDE) constrained optimization problems. The proposed work extends the recently introduced bi-level variational quantum PDE constrained optimization (BVQPCO) framework for linear PDE to a nonlinear setting by leveraging Carleman linearization (CL). CL framework allows one to transform a system of polynomial ordinary differential equations (ODE), i,e. ODE with polynomial vector field, into an system of infinite but linear system of ODE. For instance, such polynomial ODEs naturally arise when the PDE are semi-discretized in the spatial dimensions. By truncating the CL system to a finite order, one obtains a finite system of linear ODE to which the linear BVQPCO framework can be applied. In particular, the finite system of linear ODE is discretized in time and embedded as a system of linear equations. The variational quantum linear solver (VQLS) is used to solve the linear system for given optimization parameters, and evaluate the design cost/objective function, and a classical black box optimizer is used to select next set of parameter values based on this evaluated cost. We present detailed computational error and complexity analysis and prove that under suitable assumptions, our proposed framework can provide potential advantage over classical techniques. We implement our framework using the PennyLane library and apply it to solve inverse Burgers' problem. We also explore an alternative tensor product decomposition which exploits the sparsity/structure of linear system arising from PDE discretization to facilitate the computation of VQLS cost functions.