Parallelizing process model integration for model predictive control through oracle design and analysis for a Grover’s algorithm-inspired optimization strategy
Kip Nieman, Helen Durand, Saahil Patel, Daniel Koch, Paul M. Alsing
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.
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.
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.
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.
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.
This paper introduces a comprehensive quantum calculator using Qiskit that integrates quantum arithmetic and logic functions into a single calculator. The calculator relies on quantum Fourier transforms (QFT), controlled-phase gates, and quantum modular arithmetic to perform basic arithmetic operations such as addition, subtraction, and multiplication. These arithmetic techniques used quantum parallelism and interference to perform calculations more efficiently, particularly for tasks involving large numerical inputs. Numerical values are stored in qubit registers and processed using QFT-based circuits, which enable efficient and reversible mathematical computations. In addition to arithmetic operations, the calculator also implements quantum logic algorithms, including Grover’s Search and the Deutsch-Jozsa algorithm. Grover’s algorithm is an unstructured search algorithm that operates by constructing an oracle and iteratively amplifying the probability amplitude of the target solution. This amplitude amplification significantly increases the likelihood of measuring the desired outcome. Due to this property, Grover’s algorithm has practical relevance in applications such as database searching and solving combinatorial optimization problems. The Deutsch-Jozsa algorithm determines whether or not a certain Boolean function is constant or balanced and offers an exponential speed up compared to classical methods for this class of problem. A key fundamental function of the calculator is the ability to generate hybrid quantum circuits that support both arithmetic and logical functions. That allows the exploration of simple as well as complex problem-solving strategies that take advantage of both quantum arithmetic and decision-making algorithms. For instance, combining Grover’s amplification with function characterizations from the Deutsch-Jozsa algorithm provides not only interesting circuit configurations but also adds value to the possibilities of the algorithms. This research offers a modular idea and allows exploring the fundamental principles of quantum computing generally speaking and will be a valuable resource for educational as well as research purposes in the rapidly growing field of quantum algorithm development.
The application of Model Predictive Control (MPC) to quantum systems faces fundamental challenges arising from measurement-induced disturbances and the probabilistic nature of quantum dynamics. Unlike classical nonlinear systems, quantum measurements inherently disturb the system state, making feedback control intrinsically model-dependent and stochastic. Within the framework of Positive Operator-Valued Measures (POVMs), this thesis focuses on strong quantum measurements, where each observation induces a complete state collapse on the system. When uncertainty is introduced, the measurement process itself becomes a dominant source of disturbance. To address this, a lower bound on the measurement uncertainty is imposed, quantified as the predicted deviation between the measured quantum state and the nominal MPC trajectory, ensuring that the quantum state can be steered along the designed MPC trajectory while maintaining physical realizability and stability. The thesis further develops a stochastic MPC (SMPC) formulation based on quantum filtering equations, describing the conditional evolution of quantum states under continuous observation. A theoretical equivalence between the proposed SMPC scheme and an optimal control problem is established, providing a tractable computational approach for feedback design in quantum stochastic systems. Finally, to address the limitations imposed by explicit identification of quantum systems in complex quantum systems, a data-driven MPC framework is introduced. Using locally observable dynamics and trajectory data, a data-driven MPC formulation is introduced to realize predictive quantum control without explicit system identification. This framework lays the foundation for measurement-aware and physically consistent quantum MPC.
This article analyzes mathematical models of quantum parallelism based on the transfer of a quantum system into a superposition state. The principles of quantum parallelism and its mathematical foundations are substantiated. To demonstrate the quantum advantage, mathematical models of fundamental quantum algorithms Deutsch-Joža and Grover are analyzed, which illustrate the efficiency of quantum computing compared to classical methods. The Deutsch-Joshy algorithm is considered, which generalizes the problem of determining the type of a binary function by extending it to a set of arguments written in a quantum register. This algorithm allows determining whether a function is constant or balanced in a single quantum call, while in classical calculations this requires exponential time. It is noted that although the Deutsch-Joshy algorithm demonstrates the possibilities of parallel quantum computing, it is of mainly theoretical importance. Grover’s algorithm implements quantum parallel computing for the practical problem of searching for an element in an unordered database. It is based on iterative amplification of the amplitude of the state corresponding to the searched element and provides quadratic speedup compared to classical algorithms. In addition to database search, Grover’s algorithm can be adapted to solve other optimization problems. The work of the considered algorithms is illustrated by numerical examples, which simplifies the understanding of their principles and contributes to the further methodological development of parallel quantum computing. The article also outlines the advantages of quantum parallelism over classical computing and identifies the prospects for the application of parallel quantum algorithms.