Investigating an amplitude amplification-based optimization algorithm for model predictive control

The potential for greater algorithmic efficiency for some problems on quantum computers compared to classical computers is appealing in many fields including, for example, the process systems engineering field. While quantum algorithms have been studied for a variety of applications related to optimization, molecular modeling, and machine learning, there remain many applications in process systems engineering, including process control, where it is not clear how quantum computing algorithms would be beneficial. One idea for attempting to understand when a quantum algorithm might provide benefits for control is to start with algorithms that would be expected to benefit “similar” problems (e.g., optimization problems) and to see if controllers can be implemented within those algorithmic frameworks. Therefore, in this work, we study the use of a quantum computing algorithm related to Grover’s algorithm, which is an amplitude amplification strategy that can search an unordered list with improved efficiency compared to a classical algorithm for the task. It has been extended to perform a search for optimal paths over a graph. Given its potential utility for search and optimization, this is an example of an algorithm where we might wonder if it could be adjusted or used to provide speed-ups for large control problems if the controller could function within this algorithmic framework. This work provides the first steps toward attempting to address this question by investigating how optimization-based control problems would fit into this framework. A process described by $\dot{x}=x+u$ is considered as a test case. The modified Grover’s algorithm requires the optimization problem to be mapped into quantum gates. We discuss ideas for attempting to represent an optimization-based controller known as model predictive control (MPC) in the modified Grover’s algorithm framework. We test how various parameters of the control and quantum algorithm designs, including fundamental parameters in MPC such as the number of sampling periods and length of the sampling periods, impact the success of using the quantum algorithm for the MPC. We provide analyses regarding why the results are what they are to give perspective on how quantum computing algorithms work and intersect with engineering problems.