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arXiv:2207.09181 · IEEE International Conference on Quantum Computing and Engineering (QCE)

Quantum topology optimization of ground structures for near-term devices

Yuki Sato, Ruho Kondo, Satoshi Koide, Seiji Kajita

Published Sep 17, 2023• 11 citations
arXivHTMLPDF

Abstract

Product development processes frequently use numerical simulations and mathematical programming techniques. Structural optimization, especially topology optimization, is one of the most promising techniques for generating insightful design choices. Topology optimization for discrete structures, such as truss structures, can be formulated as a combinatorial optimization problem, where the combination of the existence or absence of the material is optimized. In this study, we examine the usage of quantum computers as a potential solution to discrete topology optimization problems. The proposed method consists of two variational quantum algorithms (VQAs); the first solves the state equilibrium equation for all possible structures, and the second amplifies the probability of the optimal structure in quantum superposition. Several experiments, including a real device experiment, show that the proposed method successfully obtained the optimal structure. These findings suggest that quantum computers could be a potential tool for solving topology optimization problems.

DOI
10.1109/qce57702.2023.00027

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Product development processes frequently use numerical simulations and mathematical programming techniques. Structural optimization, especially topology optimization, is one of the most promising techniques for generating insightful design choices. Topology optimization for discrete structures, such as truss structures, can be formulated as a combinatorial optimization problem, where the combination of the existence or absence of the material is optimized. In this study, we examine the usage of quantum computers as a potential solution to discrete topology optimization problems. The proposed method consists of two variational quantum algorithms (VQAs); the first solves the state equilibrium equation for all possible structures, and the second amplifies the probability of the optimal structure in quantum superposition. Several experiments, including a real device experiment, show that the proposed method successfully obtained the optimal structure. These findings suggest that quantum computers could be a potential tool for solving topology optimization problems.
Quantum scientific computing has emerged as a new paradigm to solve difficult scientific computing problems on quantum computers. One of them is topology optimization, which is computationally expensive because the combinatorial optimization problem and partial differential equations need to be solved simultaneously. In this paper, we propose a novel variational quantum algorithm for topology optimization through quantum entanglement. Two quantum registers are used to encode the optimal configurations and the solutions to physical constraints, respectively. The tasks of finding the optimal material configuration and solving the physical constraints are performed simultaneously in a single loop. A constraint encoding scheme is also proposed to incorporate volume and topology constraints in optimization. The gate complexity of the proposed quantum algorithm is analyzed. The theoretical lower bound of the success probability is obtained. The algorithm is demonstrated with compliance minimization problems including truss structures and Messerschmitt-Bölkow-Blohm beams.
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.
Abstract One of the challenging scientific computing problems is topology optimization (TO), where the two tasks of searching through the combinatorially complex configurations and solving the constraints of partial differential equations need to be done simultaneously. In this paper, a novel variational quantum algorithm for constrained TO is proposed, which allows for the single-loop parallel search for the optimal configuration that also satisfies the physical constraints. The optimal configurations and the solutions to physical constraints are encoded with two separate quantum registers. A constraint encoding scheme is also proposed to incorporate volume and connectivity constraints in optimization. The gate complexity of the proposed quantum algorithm is analyzed. The algorithm is demonstrated with compliance minimization problems including truss structures and Messerschmitt–Bölkow–Blohm beams.
Quantum computing is a new form of computational technology, which can potentially be used to solve certain problems faster than is possible using classical computers. For this reason, there is an industry drive to develop early quantum computing applications. In this thesis, an overview of quantum computing technologies is provided, along with a practical discussion of the Traveling Salesman Problem, making use of the D-Wave quantum annealer. Subsequently, the main objective of the thesis can be investigated, which is to explore how quantum computing can be used to aid in solving structural optimization problems. Two methods are developed with which simple 2-dimensional truss systems can be optimized using the D-Wave quantum annealer. The methods aim to find the most lightweight choices for the truss cross-sectional areas while complying with material limit stress constraints. The first method directly casts such an optimization problem into a QUBO format. However, due to difficulties with formulating the stress constraint, this method was found to produce a trivial optimization problem. The second method attempts to symbolically solve a truss finite-element problem, using the resulting symbolic expressions to set up an optimization objective function. Although these objective functions are confirmed to work via classical brute-force analysis, the quantum annealer is shown to have difficulty finding the global optimum solution for truss systems with three or more elements. These results indicate that it is not currently beneficial to use quantum annealing for these structural optimization problems. Nevertheless, some improvements to the method for setting up the objective functions are suggested. The next generation of quantum annealers is expected to perform better for these practical applications, potentially becoming a useful tool in the engineering toolbox.
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.
Solving combinatorial optimization problems using variational quantum algorithms (VQAs) might be a promise application in the NISQ era. However, the limited trainability of VQAs could hinder their scalability to large problem sizes. In this paper, we improve the trainability of variational quantum eigensolver (VQE) by utilizing convex interpolation to solve portfolio optimization. Based on convex interpolation, the location of the ground state can be evaluated by learning the property of a small subset of basis states in the Hilbert space. This enlightens naturally the proposals of the strategies of close-to-solution initialization, regular cost function landscape, and recursive ansatz equilibrium partition. The successfully implementation of a 404040-qubit experiment using only 101010 superconducting qubits demonstrates the effectiveness of our proposals. Furthermore, the quantum inspiration has also spurred the development of a prototype greedy algorithm. Extensive numerical simulations indicate that the hybridization of VQE and greedy algorithms achieves a mutual complementarity, combining the advantages of both global and local optimization methods. Our proposals can be extended to improve the trainability for solving other large-scale combinatorial optimization problems that are widely used in real applications, paving the way to unleash quantum advantages of NISQ computers in the near future.