(176c) On Probabilistic Input Selection: Utilizing a Quantum Algorithm in Selecting Inputs for Lyapunov-Based Economic Model Predictive Control Formulated As a Look-up Table

Recent years have seen an increasing interest in the potential uses of quantum computing [1, 2]. For example, [2] and [3] developed algorithms for quantum computing in the context of optimization and machine learning from a chemical process systems perspective. At this time, however, much of the potential of quantum computing, both within chemical engineering and in engineering in general, remains unclear. This is due to the physics which underlies quantum computing calculations which necessitate specialized algorithms to attempt to out-do classical algorithms for the same engineering problems. Furthermore, the inherent uncertainty in quantum mechanics can introduce uncertainty in the results of quantum computing calculations. For example, Grover’s Search Algorithm, an algorithm for searching through a set of strings of binary digits, can yield a certain result only in probability [1, 4]. This type of behavior of quantum computers raises interesting questions for chemical engineering computations that are typically assumed to be performed “correctly,” and where correctness can matter from a plant safety standpoint. For example, it is interesting to consider how quantum computing algorithms with probabilistic properties might interact with control design and theory.

As a preliminary step toward addressing these considerations, this talk will focus on the use of a model predictive control algorithm, where the state measurement to controller output relationship is formulated as a lookup table (despite the computational complexity of this; i.e., this preliminary study on the potential interactions between probabilistic control actions generated by a quantum computer and closed-loop stability considerations will neglect questions regarding the practicality of model predictive control formulated as a look-up table). Due to its strong stability and feasibility properties, we will select the specific model predictive control algorithm known as Lyapunov-based economic model predictive control (LEMPC) [5]. We will formulate the LEMPC problem as a search problem – given a state measurement, the corresponding control action must be located in the lookup table. Formulating the LEMPC problem in this context enables us to consider that the search for the correct control action from the lookup table is performed on a quantum computer, facilitated by Grover’s Search Algorithm. Though the use of Grover’s Search Algorithm in this context may not be necessary to locate the appropriate control action on a quantum computer, the use of this algorithm as part of the strategy will enable the probabilistic nature of a quantum computation to interact with the determination and application of an optimal input to the process.

The initial step toward developing this LEMPC formulated as a look-up table for a quantum computer is to consider that both the state measurement and the actuator output can only be known to fixed precision. This enables both to be represented as finite strings of binary digits of fixed length for application of Grover’s Search Algorithm. Initially, we will review the stability conditions required for lookup table LEMPC with fixed precision for the state measurements and actuator outputs in the case that the control actions are computed on a classical computer. Then, we will analyze the case that the control actions are selected from the lookup table using Grover’s Search Algorithm. To treat the stochasticity, we will consider that the inputs are bounded, so that the (potentially undesired) input cannot be arbitrarily bad. We will explore how the selection of the input using these probabilistic computing methods impacts closed-loop stability considerations, and will also discuss the manner in which noise in the quantum circuits themselves [6] can play a role in the stability results. Finally, we will discuss the potential of other available quantum algorithms for application in a control context.

REFERENCES:

[1] M. A. Nielsen and I. Chuang. Quantum Computation and Quantum Information, 10th Anniversary Edition. Cambridge University Press, New York, 2010.

[2] A. Ajagekar, F. You. “Quantum Computing Assisted Deep Learning for Fault Detection and Diagnosis in Industrial Process Systems.” Computers & Chemical Engineering. Volume 143, 107119, 2020.

[3] A. Ajagekar, T. Humble, F. You. “Quantum Computing based Hybrid Solution Strategies for Large-scale Discrete-Continuous Optimization Problems.” Computers & Chemical Engineering. Volume 132, 106630, 2020.

[4] N. S. Yanofsky and M. A. Mannucci. Quantum Computing for Computer Scientists. Cambridge University Press, New York, NY, 2008.

[5] M. Ellis, H. Durand, P. D. Christofides. “A tutorial review of economic model predictive control methods.” Journal of Process Control. Volume 24, Issue 8, Pages 1156-1178, 2014.

[6] D. Koch, B. Martin, S. Patel, L. Wessing, and P. M. Alsing. “Demonstrating NISQ era challenges in algorithm design on IBM’s 20 qubit quantum computer.” AIP Advances, Volume 10, 095101, 2020.