Variational Quantum Framework for Partial Differential Equation Constrained Optimization

We present a novel variational quantum framework for linear partial differential equation (PDE) constrained optimization problems. Such problems arise in many scientific and engineering domains. For instance, in aerodynamics, the PDE constraints are the conservation laws such as momentum, mass and energy balance, the design variables are vehicle shape parameters and material properties, and the objective could be to minimize the effect of transient heat loads on the vehicle or to maximize the lift-to-drag ratio. The proposed framework utilizes the variational quantum linear system (VQLS) algorithm and a black box optimizer as its two main building blocks. VQLS is used to solve the linear system, arising from the discretization of the PDE constraints for given design parameters, and evaluate the design cost/objective function. The black box optimizer is used to select next set of parameter values based on this evaluated cost, leading to nested bi-level optimization structure within a hybrid classical-quantum setting. We present detailed computational error and complexity analysis to highlight the potential advantages of our proposed framework over classical techniques. We implement our framework using the PennyLane library, apply it to a heat transfer optimization problem, and present simulation results using Bayesian optimization as the black box optimizer. We also demonstrate that by using an alternative tensor product decomposition which better exploits the sparsity and structure of linear systems arising from PDE discretizations, one can substantially overcome the key computational bottleneck in VQLS arising from commonly employed Pauli basis for the linear combination of unitary (LCU) decomposition step within VQLS.