  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.
