For these reasons, quantum computing is an attractive alternative for exploration. Quantum algorithms are appealing because they have been mathematically demonstrated to outperform classical algorithms in certain cases. In chemical engineering and related fields, quantum algorithms have been studied for a wide variety of applications including computational chemistry tasks such as the determination of reaction mechanisms [6] and thermodynamic properties [7], the study of biological compounds such as proteins [8] and peptides [9], energy control and management [10, 11], routing problems and logistics optimization [12, 13], and process and product design [13]. However, to date no quantum computer has outperformed the best classical solvers in a fair head-to-head comparison involving an industry-scale problem [2]. Challenges remain in both hardware and algorithm development, but the potential for future use is promising given the rate at which new technologies and algorithms are advancing.
In this work, we narrow our focus to a specific class of optimization called quadratic unconstrained binary optimization (QUBO) problems. These problems involve objective functions consisting of weighted linear and quadratic terms, where the decision variables are binary. Several quantum algorithms have shown promise for solving QUBOs, including the quantum approximate optimization algorithm (QAOA) [14, 15] and the variational quantum eigensolver (VQE) [15]. We focus on a third potential candidate known as amplitude amplification, which is a generalization of Grover's search algorithm [16]. Amplitude amplification begins with an equal superposition state, representing equal uncertainty among all 2^N solutions, and consists of two repeated steps. The first step involves the application of an operator called the 'oracle', encoding the weight values of an objective function as phases. The second involves a 'diffusion' operator, which performs a mathematical process called 'inversion about the mean.' Repeated applications of the oracle and diffusion operators lead to an increased probability that the measurement of the final quantum state will yield the decision variable values corresponding to the maximum objective function value. The scaling of the problem is favorable because the oracle can encode 2^N solutions of the objective function using only N qubits, but implementation of the algorithm on today’s hardware remains a challenge. There has been a great deal of mathematical and theoretical work investigating the inner workings of the algorithm [16, 17], but there are notably fewer studies investigating amplitude amplification for real-world problems.
Previous works have identified several important factors for determining whether the algorithm is successful: the distribution of objective function values, the number of iterations of the oracle and diffusion operator, and the value of a scaling parameter within the oracle. It is not always clear what the distribution of solutions for a real-world problem will be, which in turn is necessary for determining the optimal number of iterations and scaling parameter for the algorithm. These issues represent fundamental challenges to applying amplitude amplification for industry-scale engineering applications. This work seeks to address these challenges through the creation and study of a simulation testbed. The simulation is representative of a manufacturing routing control problem, where items or parts need to be delivered throughout an assembly area. The simulation assumes a grid of conveyor belts that can move objects along known paths, but with limited direction of movement allowed over each control period. Each binary decision variable represents an intersection of conveyor belts, where the values of zero or one determine north-south versus east-west movement. The challenge is to optimally choose the direction at each intersection in order to maximize a QUBO objective function representing total distance of movement across all objects.
In this work, we begin by giving a general overview of quantum computing with a few examples of the mathematical framework utilized in quantum algorithms. We then describe the quantum amplitude amplification algorithm in detail, the oracle and diffusion operators, the way in which optimal values increase in probability through geometrical transformations in the complex plane, how to construct an oracle encoding QUBO problems, and finally the challenges that remain for solving real-world applications. Next we discuss the details of the QUBO problem created from conveyor belt control simulation, and methods for predicting near optimal values for the quantum algorithm. Finally, we present simulation results demonstrating performance of the control system and conclude with future research directions such as strategies for expanding the algorithm encompass more classes of problems.
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