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 ẋ=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.
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 ẋ=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.
Quantum computers, which utilize quantum states called ‘qubits’ to process information, are becoming of increased interest in a variety of fields because they have the potential to outperform classical computers in certain situations. However, many challenges and hurdles remain, including the development of quantum algorithms that offer a speedup and can be applied to practical problems (current quantum algorithms that offer any sort of speedup are often only applicable in specific and sometimes contrived circumstances). Our previous works have studied the use of quantum computers and quantum algorithms for process control applications. Our prior work evaluated the use of a gate-based quantum amplitude amplification algorithm when applied to a model predictive control optimization problem, specifically evaluating a linear systems example. The results suggested that the algorithm may be suited for the specific linear problem studied (meaning that there is a high probability of obtaining the desired result when measuring the quantum state after the algorithm has been applied). However, the results cannot be generalized to a wider class of systems or to systems containing nonlinearities. To begin to gain an understanding of how the amplitude amplification algorithm might be generalized, this work evaluates the use of the algorithm for the optimization-based control for a nonlinear dynamic system (specifically, a continuous stirred tank reactor). We seek to understand aspects of the problem, such as the development of a solution space, the effects of changing process and control parameters, and the operation of the amplitude amplification algorithm itself. The results demonstrate the challenges that still exist, as well as a method involving inverse sampling transformations to potentially address these issues.
We introduce a new hybrid quantum-classical algorithm inspired by an advanced control strategy known as model predictive control (MPC). This algorithm unifies the optimization-based design of variational quantum algorithms (VQAs) with the feedback-based design of feedback-based quantum algorithms (FQAs). Variational circuit parameters are optimized using a layer-wise receding horizon strategy, where observable measurements after every layer initialize a classically simulated dynamic model used to predict quantum state evolution and optimize over future parameterized gates. This hybrid algorithm can be used for applications such as ground state preparation and approximate combinatorial optimization, and presents an ideal use case for the integration of quantum computers with high-performance computing, where the latter resource can be used to increase the scale and efficiency of the predictions critical to MPC. We show through mathematical proof and numerical evidence that the MPC-based algorithm can be guaranteed to at least match the performance of FQAs. Through simulations on Max-Cut problems and a two-dimensional transverse-field Ising model, we demonstrate that relaxed implementations of the MPC-based algorithm can also provide improved performance in practice compared to an FQA.
In model predictive control (MPC), a process dynamic model is utilized to make predictions of the value of the objective function and constraints throughout a prediction horizon. In one method of solving this problem, the time required to find the optimal values of the decision variables depends on the time required to perform the arithmetic operations involved in computing the model predictions. Methods for attempting to reduce the computation time of an MPC could then include developing approximate (reduced-order or data-driven) models for a system that take less time to solve, or to parallelize the computations using, for example, multiple cores or CPU’s. However, an observation in all of these cases is that the values of the process states across the prediction horizon are not the values returned by the optimization problem; the manipulated input trajectory is the desired decision variable. An optimization strategy that cannot explicitly return the process states but can generate some representation of them that then leads to computation of the desired process input would thus be suitable for MPC. Quantum computers achieve their parallelism through creating values that cannot all be returned. They can then operate on this set of values to return a number that is meaningful with respect to that set of values that could not all be returned. Motivated by this, we wish to investigate an idea for utilizing quantum parallelism in developing a representation of an objective function that depends on the solution of a process dynamic model, but then only returning the control actions that minimize the objective function value dependent on those solutions. To do this, several steps are necessary. The first is to locate a quantum algorithm which has the desired characteristics for achieving the goals. In this work, we perform these steps using an amplitude amplification strategy based on Grover’s algorithm on a quantum computer. The second is to analyze the algorithm with respect to its ability to translate across problems in the MPC domain, with respect to both its ability to handle nonlinear systems and to handle a variety of different structures of the set of all possible objective function values given the allowable values of the decision variables. We thus evaluate the benefits and limitations of the algorithm from this perspective. We do not wish to imply that this algorithm is more computationally-tractable for use with MPC than classical optimization techniques traditionally applied in an MPC context. Rather, we wish to understand the manner in which such an algorithm would be designed and when it is appropriate for MPC problems (in the sense of returning the correct answers to the optimization problem), as a step toward better understanding the interactions of the quantum properties of quantum algorithms with control goals. We also see this as forming an important first step in algorithm design/analysis, which can then translate to future works in comparing this technique computationally with classical “competitor” algorithms to guide further work in searching for relevant quantum algorithms for control applications.
This paper investigates the application of quantum-enhanced constraint optimization to robotics, specifically focusing on dynamic adjustment of robot control parameters to maximize task performance. Traditional optimization methods often struggle with complex, dynamic environments, and quantum computing offers the potential for significant speed-ups in solving optimization problems. We propose a novel algorithm leveraging a quantum-inspired optimization technique – specifically, a modified version of Grover's algorithm – to rapidly identify optimal control parameters. The core claim is that this algorithm can dramatically improve the efficiency and responsiveness of robotic systems, particularly in tasks requiring real-time adaptation. This work demonstrates the feasibility of using quantum mechanics principles to accelerate optimization processes, ultimately enabling more sophisticated and adaptable robotic control.
Quantum search is one of the most important algorithmic developments in quantum computing because it demonstrates how quantum principles can improve the efficiency of solving unstructured search problems. Grover’s algorithm provides a quadratic speed-up over classical linear search by reducing query complexity from O(N) to O(√N), where N represents the size of the search space. This review examines the principles, performance, and applications of Grover’s algorithm, with emphasis on its theoretical foundation and practical relevance. The discussion covers the roles of superposition, quantum interference, oracle operations, diffusion operators, and amplitude amplification in increasing the probability of measuring a desired solution. The manuscript also reviews the computational performance of quantum search in comparison with classical search methods and considers its relevance to database searching, combinatorial optimisation, cryptography, machine learning, bioinformatics, quantum chemistry, and materials science. The review highlights that Grover’s algorithm is not limited to database retrieval, as many computationally intensive problems can be reformulated as search or optimisation tasks. However, practical implementation remains constrained by current quantum hardware limitations, including decoherence, quantum noise, limited qubit availability, circuit depth, and the difficulty of designing efficient oracles. Despite these challenges, continuing progress in quantum hardware, error correction, cloud-based quantum platforms, and hybrid quantum-classical architectures suggests that quantum search will remain an important area of research. Overall, Grover’s algorithm represents a foundational quantum algorithm with significant theoretical value and potential long-term application in computational domains involving large and unstructured search spaces.
The K-SAT problem is a fundamental challenge in computational theory, widely used in artificial intelligence, cryptography, and optimization. Classical algorithms struggle with exponential time complexity, making them inefficient for large-scale instances. In this research, we propose an efficient quantum computing approach based on Grover’s algorithm to enhance the search process for satisfiable solutions in K-SAT problems. Our implementation leverages quantum superposition and amplitude amplification to explore multiple possible solutions simultaneously, reducing the search complexity from O (2ⁿ) in classical methods to O(√2ⁿ) in quantum computing. We design a quantum oracle that encodes the K-SAT clauses and integrates it into Grover’s iterative search framework. The performance is evaluated through Qiskit simulations, demonstrating a significant improvement in search efficiency compared to classical brute-force techniques. The results highlight the potential of quantum algorithms in solving complex combinatorial problems with enhanced speed and accuracy. This study contributes to the development of quantum-accelerated optimization methods, paving the way for real-world applications in machine learning, cryptanalysis, and large-scale data processing.