Researchers Optimize Truss Structures Using Quantum Algorithm with Fixed Linear Ramp Schedule

Computational engineering researchers have established a hybrid optimization framework that pairs classical structural analysis with the Quantum Approximate Optimization Algorithm to determine optimal truss member sizing while bypassing the computational bottleneck of iterative parameter tuning. In a study published in Advanced Modeling and Simulation in Engineering Sciences, the team implemented a fixed linear ramp schedule, commonly abbreviated as FLRS, within the variational quantum circuit of the Quantum Approximate Optimization Algorithm, or QAOA. By establishing the variational parameters analytically through a predetermined linear progression rather than relying on an external classical optimizer at every cycle, the proposed framework avoids the notorious barren plateau problem and achieves consistent structural convergence across multi-member benchmark structures.

Truss structures represent fundamental assemblies in structural design where engineers seek to minimize compliance, which corresponds to maximizing overall structural stiffness, under specific external loads while strictly adhering to volume or weight constraints. Classical approaches have historically utilized gradient-based strategies, genetic algorithms, or evolutionary structural optimization, often combined with parallel computing or dimensionality reduction. However, traditional algorithmic enhancements face performance plateaus due to computational bottlenecks and saturation in classical hardware development. While quantum computing has emerged as a promising alternative capable of exploring candidate solutions in superposition across a Hilbert space, mapping continuous structural member dimensions directly to discrete quantum states has remained a major challenge in computational mechanics.

Previous investigations into applying gate-based universal quantum computing to structural problems formulated member sizing either as binary inclusion-exclusion ground structure choices or as one-hot encoded variables. In binary inclusion models, entire structural members are either retained or deleted, creating an exponential growth in objective function complexity as the number of rods increases. Conversely, one-hot encoding assigns discrete cross-sectional area choices using individual qubits per candidate value, preserving quadratic cost formulations at the expense of requiring substantial numbers of qubits. Furthermore, traditional variational algorithms require an inner iterative loop where classical numerical optimizers continually refine circuit angles, creating an expensive double-iteration process that severely degrades computational efficiency.

Instead of restricting a member to binary presence or allocating extensive qubit arrays, the framework discretizes each updater through an on-off encoding scheme using only two qubits per rod. In on-off encoding, candidate values are expressed as linear combinations of predefined weights, allowing m qubits to represent two to the power of m options without requiring restrictive one-hot constraints. With two qubits assigned to each rod, four operational updater multipliers are defined: a negligible value epsilon between one and two times ten to the negative tenth power for rod removal, one-tenth for area reduction, one for maintaining the current dimension, and one and one-tenth for area reinforcement.

The physical iteration process couples a classical finite element solver with a simulated quantum processor using the PennyLane quantum programming platform. In each optimization cycle, the classical processor evaluates the static equilibrium equations of the current truss configuration to compute global nodal displacements and internal strain energy distributions. Rods experiencing elevated strain energy naturally benefit from an increased cross-sectional area, whereas under-stressed members can have their cross sections reduced or eliminated. To manage overall material consumption, a slack variable representing potential volume violations is also discretized using two additional qubits, normalizing the volume constraint penalty into a unified quadratic objective function suitable for quantum cost Hamiltonians.

The variational quantum circuit executes alternating layers of problem-specific phase-separating operators and transversal mixer operators composed of single-qubit Pauli-X rotations. In contrast, the implementation of FLRS prescribes the mixer angles beta to decrease steadily across the circuit layers while the phase angles gamma increase steadily according to a fixed linear ramp. This predetermined parameterization relies on the physical analogy between deep QAOA circuits and adiabatic quantum evolution, completely eliminating classical optimization overhead for variational parameters.

The effectiveness of the schedule was evaluated through numerical experiments on a benchmark six-member two-dimensional truss subjected to a vertical downward point load of one hundred kilonewtons. Using a six-layer circuit requiring twelve variational parameters, the authors compared FLRS directly against the widely adopted COBYLA and Bayesian optimization methods across ten independent optimization executions. While COBYLA and Bayesian routines exhibited significant dispersion and failed to converge reliably due to initial sampling sensitivities and barren plateaus, the FLRS framework achieved low mean absolute percentage error relative to reference solutions calculated with classical optimality criteria methods. Crucially, runtime measurements confirmed that FLRS executed the initial structural updating step substantially faster than classical optimizers by removing the parameter-tuning search loop.

Following validation on the six-member system, the researchers evaluated an eleven-member two-dimensional truss structure utilizing an eight-layer QAOA circuit executed with twenty-four total qubits, representing twenty-two member qubits and two slack variable qubits. The optimization progressed through successive iterations, successfully identifying and eliminating redundant members while progressively redistributing material into load-carrying pathways. The resulting structural member arrangement converged to a topology identical to that generated by classical optimality criteria techniques. Minor structural volume fluctuations around the nominal target constraint were observed during convergence, which the authors attributed to the coarse discretization inherent to two-qubit encoding and the numerical residual epsilon retained on deleted rods to preserve matrix stability.

To enhance computational efficiency when scaling to larger structural systems, the authors introduced an early stopping update heuristic. Under this protocol, when a member cross section decreases below one percent of its initial value, whether through immediate selection of the negligible multiplier epsilon or through repeated downscaling steps, the framework freezes that rod and removes its associated qubits from subsequent circuit iterations. Because the state-space and query complexity of combinatorial quantum optimization depend directly on active qubit counts, pruning inactive elements exponentially curtails circuit simulation requirements and frees quantum resources during intermediate design iterations.

The authors note specific constraints and trade-offs within the current implementation. The penalty weighting parameter lambda that balances structural compliance minimization against volume constraint satisfaction was determined empirically through trial and error. Setting this multiplier too low permits structural volume violations, whereas excessive penalty weighting forces the algorithm to prioritize constraint satisfaction at the expense of structural stiffness, risking inadvertent removal of critical load-bearing members. Because the research relied on ideal quantum simulation within PennyLane rather than physical noisy intermediate-scale quantum processors, full implementation on physical gate-based quantum computers will require addressing quantum gate fidelity, noise mitigation, and expanded physical qubit counts as quantum hardware matures.

In both test cases, the simulated trusses featured rods with lengths of 1.0 meter or the square root of two meters, a Young’s modulus of 2 × 10^11 newtons per square meter, and an initial cross-sectional area of 0.5 square meters. These specifications established baseline structural volumes of 3.414 square meters for the six-rod system and 6.328 square meters for the eleven-rod system. To verify algorithm fidelity, the geometry incorporated unloaded vertical members on the far left that transmitted no force under the applied 100-kilonewton load; in both cases, the optimization properly eliminated these inactive rods during early design iterations. Because quantum circuit outputs cannot yield expectation values in a single shot, the researchers performed 100,000 measurements per circuit execution to suppress statistical sampling uncertainty, identifying the most frequently occurring bitstring to guide structural updates.

The authors also evaluated the theoretical time complexity of their methodology against competing quantum structural formulations. Previous ground structure optimization methods required classical processors to evaluate extensive polynomial expansions containing two to the power of the qubit count terms, producing severe computational bottlenecks despite minimal qubit requirements. Other non-iterative approaches utilizing one-hot encoding avoided exponential polynomial expansion but demanded four or more qubits per rod alongside iterative classical optimizers like Adam, multiplying both qubit demands and execution time. In contrast, the quadratic cost Hamiltonian of the authors’ framework matches the structure of the standard Max-Cut problem. By pairing fixed linear ramp parameter schedules with the early stopping heuristic that systematically prunes neglected members, the iterative framework scales with an estimated time complexity proportional to circuit depth, active iterations, and member count, governed by a scaling factor strictly bounded between one and two.

Looking toward future scalability, the researchers analyzed theoretical qubit demands for larger systems, noting that quantum simulation beyond fifty qubits becomes computationally prohibitive on classical computers. Because member updaters consume qubits linearly, optimizing a fifty-member structure would require one hundred design qubits. Meanwhile, expressiveness for the slack variable plateaus rapidly; allocating ten qubits provides 1,024 discrete options, eliminating the need to expand constraint qubits further and bounding the total requirement for fifty members between 105 and 110 qubits. The authors also noted that while the fixed linear ramp schedule effectively circumvents classical parameter searches, its heuristic nature means ramp angles may not prove universally optimal across all engineering geometries. They suggested that ramp schedules could serve as robust initial configurations, which classical optimizers could then refine to deliver enhanced precision in complex structural applications.

Subject of Research: Quantum Approximate Optimization Algorithm for structural truss optimization

Article Title: An efficient Quantum Approximate Optimization Algorithm with fixed linear ramp schedule for truss structure optimization

Article References: Xiao, J., Sukulthanasorn, N., Nomura, R., Moriguchi, S., & Terada, K. (2025). An efficient Quantum Approximate Optimization Algorithm with fixed linear ramp schedule for truss structure optimization. Advanced Modeling and Simulation in Engineering Sciences, 12(1), Article 14. https://doi.org/10.1186/s40323-025-00298-2

Image Credits: AI Generated

DOI: 10.1186/s40323-025-00298-2

Keywords: quantum computing, structural optimization, truss structures, Quantum Approximate Optimization Algorithm, QAOA, fixed linear ramp schedule, finite element method, on-off encoding, computational mechanics, variational quantum circuits, engineering design, compliance minimization, quantum simulation, PennyLane, qubit encoding, mechanical engineering