Abstract Despite its wide scope of applications, topology optimization faces a major challenge, which is its high computational expenses caused by the large number of design variables and the underlying physics. Because the number of possible material layouts increases exponentially, the search for the optimum becomes computationally intractable. Recently, quantum computing emerged as an alternative paradigm for solving optimization problems with potentially much higher efficiency than classical computing. The quantum superposition phenomenon allows quantum computer to perform a parallel search of many possible solutions. In this work, a two-mixer quantum approximate Bayesian optimization algorithm (TM-QABOA) is developed as a hybrid quantum-classical optimization algorithm, where Pauli-X mixers and generalized Grover mixers are applied in an alternating fashion. TM-QABOA enables a dynamic exploration-exploitation balance to improve searching efficiency. The exploration is done with the Pauli-X mixers, whereas the exploitation is enhanced by the generalized Grover mixers through amplitude amplification. Surrogate-based Bayesian optimization is used to optimize the hyperparameters of the quantum circuit, i.e., rotation angles. The algorithm can also be used to solve mixed-integer optimization problems. The feasibility of TM-QABOA to solve optimization problems is demonstrated with two simple examples. One is two-dimensional truss design, and the other is metamaterials design.
Abstract Despite its wide scope of applications, topology optimization faces a major challenge, which is its high computational expenses caused by the large number of design variables and the underlying physics. Because the number of possible material layouts increases exponentially, the search for the optimum becomes computationally intractable. Recently, quantum computing emerged as an alternative paradigm for solving optimization problems with potentially much higher efficiency than classical computing. The quantum superposition phenomenon allows quantum computer to perform a parallel search of many possible solutions. In this work, a two-mixer quantum approximate Bayesian optimization algorithm (TM-QABOA) is developed as a hybrid quantum-classical optimization algorithm, where Pauli-X mixers and generalized Grover mixers are applied in an alternating fashion. TM-QABOA enables a dynamic exploration-exploitation balance to improve searching efficiency. The exploration is done with the Pauli-X mixers, whereas the exploitation is enhanced by the generalized Grover mixers through amplitude amplification. Surrogate-based Bayesian optimization is used to optimize the hyperparameters of the quantum circuit, i.e., rotation angles. The algorithm can also be used to solve mixed-integer optimization problems. The feasibility of TM-QABOA to solve optimization problems is demonstrated with two simple examples. One is two-dimensional truss design, and the other is metamaterials design.
The searching efficiency of the quantum approximate optimization algorithm is dependent on both the classical and quantum sides of the algorithm. Recently, a quantum approximate Bayesian optimization algorithm (QABOA) that includes two mixers was developed, where surrogate-based Bayesian optimization is applied to improve the sampling efficiency of the classical optimizer. A continuous-time quantum walk mixer is used to enhance exploration, and the generalized Grover mixer is also applied to improve exploitation. In this article, an extension of the QABOA is proposed to further improve its searching efficiency. The searching efficiency is enhanced through two aspects. First, two mixers, including one for exploration and the other for exploitation, are applied in an alternating fashion. Second, uncertainty of the quantum circuit is quantified with a new quantum Matérn kernel based on the kurtosis of the basis state distribution, which increases the chance of obtaining the optimum. The proposed new two-mixer QABOA's with and without uncertainty quantification are compared with three single-mixer QABOA's on five discrete and four mixed-integer problems. The results show that the proposed two-mixer QABOA with uncertainty quantification has the best performance in efficiency and consistency for five out of the nine tested problems. The results also show that QABOA with the generalized Grover mixer performs the best among the single-mixer algorithms, thereby demonstrating the benefit of exploitation and the importance of dynamic exploration–exploitation balance in improving searching efficiency.
Bayesian Optimization (BO) is a powerful technique for optimizing black-box functions, frequently employed in areas such as machine learning hyperparameter tuning and experimental design. However, its performance can degrade significantly in high-dimensional spaces, leading to slow convergence and increased computational cost. This paper introduces a novel approach to Bayesian Optimization inspired by quantum amplitude amplification, a technique rooted in quantum computing. Our method utilizes a probabilistic representation of candidate solutions within a Hilbert space, employing amplitude amplification to systematically bias the search towards promising regions of the solution space. This approach aims to accelerate the optimization process, particularly in scenarios where traditional BO methods struggle due to the curse of dimensionality. The core claim of this work is to demonstrate a speedup in convergence compared to standard BO algorithms, leveraging the principles of amplitude amplification for enhanced exploration and exploitation. The theoretical framework is presented, alongside a detailed explanation of the algorithm's components, and the results are discussed, highlighting the potential for improved efficiency in complex optimization problems. We focus on the core mechanics of using amplitude amplification to enhance the probability of finding optimal solutions.
Abstract Topology optimization is a key methodology in engineering design for finding efficient and robust structures. Due to the enormous size of the design space, evaluating all possible configurations is typically infeasible. In this work, we present an end-to-end, fault-tolerant quantum algorithm for topology optimization that operates on an exponentially large Hilbert space representing the design space. We demonstrate the algorithm on the two-dimensional Messerschmitt-Bölkow-Blohm beam problem. By restricting design variables to binary values, we reformulate the compliance minimization task as a combinatorial satisfiability problem, solved using Grover’s algorithm. Within Grover’s oracle, the compliance is computed through the finite-element method using established quantum algorithms, including block-encoding of the stiffness matrix, quantum singular value transformation for matrix inversion, Hadamard test, and quantum amplitude estimation. The complete algorithm is implemented and validated using classical quantum-circuit simulations. A detailed complexity analysis shows that the method evaluates the compliance of exponentially many structures in quantum superposition in polynomial time. In the global search, our approach maintains Grover’s quadratic speedup compared to classical unstructured search. Overall, the proposed quantum workflow demonstrates how quantum algorithms can advance the field of computational science and engineering.
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.
This paper presents a novel approach to dynamic topology optimization, leveraging quantum computing to accelerate the process of generating and refining geometric models. Traditional topology optimization relies heavily on computationally intensive simulations, often requiring significant time and resources. We propose a quantum-enhanced algorithm that utilizes quantum algorithms to efficiently explore the geometric space, thereby reducing the computational burden and enabling the discovery of optimal designs. The core mechanism focuses on utilizing quantum state representations to represent and manipulate geometric parameters, facilitating faster convergence and more nuanced optimization. The study demonstrates the potential for significant improvements in optimization speed and accuracy compared to classical methods, particularly for complex geometries. The research highlights the promise of quantum computing to revolutionize geometric design and problem-solving.
Quantum computing has become a pivotal innovation in computational science, offering novel avenues for tackling the increasingly complex and high-dimensional optimization challenges inherent in engineering design. This paradigm shift is particularly pertinent in the domain of structural optimization, where the intricate interplay of design variables and constraints necessitates advanced computational strategies. In this vein, the gate-based variational quantum algorithm utilizes quantum superposition and entanglement to improve search efficiency in large solution spaces. This paper delves into the gate-based variational quantum algorithm for the discrete variable truss structure size optimization problem. By reformulating this optimization challenge into a quadratic, unconstrained binary optimization framework, we bridge the gap between the discrete nature of engineering optimization tasks and the quantum computational paradigm. A detailed algorithm is outlined, encompassing the translation of the truss optimization problem into the quantum problem, the initialization and iterative evolution of a quantum circuit tailored to this problem, and the integration of classical optimization techniques for parameter tuning. The proposed approach demonstrates the feasibility and potential of quantum computing to transform engineering design and optimization, with numerical experiments validating the effectiveness of the method and paving the way for future explorations in quantum-assisted engineering optimizations.