Variational Quantum Framework for Partial Differential Equation Constrained Optimization

TL;DR

This paper introduces a hybrid quantum-classical framework utilizing variational quantum algorithms to solve PDE-constrained optimization problems, demonstrating potential advantages through simulation of heat transfer optimization.

Contribution

It develops a novel variational quantum approach combining VQLS and black box optimization for PDE-constrained problems, with detailed error and complexity analysis.

Findings

Successful implementation using PennyLane library

Application to heat transfer optimization problem

Simulation results with Bayesian optimization

Abstract

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.