This study introduces a framework for learning a low-depth surrogate quantum circuit (SQC) that approximates the nonlinear, dissipative, and hence non-unitary Bhatnagar–Gross–Krook (BGK) collision operator in the lattice Boltzmann method (LBM) for the lattice. By appropriately selecting the quantum state encoding, circuit architecture, and measurement protocol, non-unitary dynamics emerge naturally within the physical population space. This approach removes the need for probabilistic algorithms relying on ancilla qubits and post-selection to reproduce dissipation, or for multiple state copies to capture nonlinearity. The SQC is designed to preserve key physical properties of the BGK operator, including mass conservation, scale equivariance, and equivariance, while momentum conservation is encouraged through penalization in the training loss. When compiled to the IBM Heron quantum processor's native gate set, assuming all-to-all qubit connectivity, the circuit requires only 724 native gates and operates locally on the velocity register, making it independent of the lattice size. The learned SQC is validated on two benchmark cases, the Taylor–Green vortex decay and the lid-driven cavity, showing accurate reproduction of vortex decay and flow recirculation. While integration of the SQC into a quantum LBM framework presently requires measurement and re-initialization at each timestep, the necessary steps towards a measurement-free formulation are outlined.

Lattice Gas Automata (LGA) is a classical method for simulating physical phenomena, including Computational Fluid Dynamics (CFD). Quantum LGA (QLGA) is the family of methods that implement LGA schemes on quantum computers. In recent years, QLGA has garnered attention from researchers thanks to its potential of efficiently modeling CFD processes by either reducing memory requirements or providing simultaneous representations of exponentially many LGA states. In this work, we introduce novel building blocks for QLGA algorithms that rely on computational basis state encodings. We address every step of the algorithm, from initial conditions to measurement, and provide detailed complexity analyses that account for all discretization choices of the system under simulation. We introduce multiple ways of instantiating initial conditions, efficient boundary condition implementations for novel geometrical patterns, a novel collision operator that models less restricted interactions than previous implementations, and quantum circuits that extract quantities of interest out of the quantum state. For each building block, we provide intuitive examples and open-source implementations of the underlying quantum circuits.

We present QLBM, a Python software package designed to facilitate the development, simulation, and analysis of Quantum Lattice Boltzmann Methods (QBMs). QLBM is a modular framework that introduces a quantum component abstraction hierarchy tailored to the implementation of novel QBMs. The framework interfaces with state-of-the-art quantum software infrastructure to enable efficient simulation and validation pipelines, and leverages novel execution and pre-processing techniques that significantly reduce the computational resources required to develop quantum circuits. We demonstrate the versatility of the software by showcasing multiple QBMs in 2D and 3D with complex boundary conditions, integrated within automated benchmarking utilities. Accompanying the source code are extensive test suites, thorough online documentation resources, analysis tools, visualization methods, and demos that aim to increase the accessibility of QBMs while encouraging reproducibility and collaboration. Program summary: Program Title: QLBM CPC Library link to program files: https://doi.org/10.17632/28hkvsg7p2.1 Developer's repository link: https://github.com/QCFD-Lab/qlbm Licensing provisions: MPL-2.0 Programming language: Python3 Supplementary material: The documentation of is available at https://qcfd-lab.github.io/qlbm/. Nature of problem: The advent of quantum algorithms for computational fluid dynamics brings with it challenges that are new to the established field of computational physics. These challenges include the lack of standardized implementations of the still nascent quantum methods, the intense computational demands of developing and simulating quantum algorithms on hardware available today, and the absence of tools that integrate novel developments into established infrastructure. Because of these current limitations, physicists and mathematicians expend superfluous resources on tasks that more mature computational physics branches have surmounted long ago. Solution method: QLBM is a software package that provides an end-to-end development environment for quantum lattice Boltzmann methods. The modular design and flexible quantum circuit library provide a base for extending and generalizing quantum algorithms. Performance enhancements exploit the paradigm of quantum computing simulations to accelerate the speed at which researchers can verify the validity of their methods. Its integration with state-of-the-art quantum computing software and visualization tools increases the algorithms' accessibility. These features allow QLBM to effectively generate, simulate, and analyze quantum circuits for 2D and 3D computational fluid dynamics problems.

Wrinkling is the phenomenon of out-of-plane deformation patterns in thin walled structures, as a result of a local compressive (internal) loads in combination with a large membrane stiffness and a small but non-zero bending stiffness. Numerical modelling typically involves thin shell formulations. As the mesh resolution depends on the wrinkle wave lengths, the analysis can become computationally expensive for shorter ones. Implicitly modelling the wrinkles using a modified kinematic or constitutive relationship based on a taut, slack or wrinkled state derived from a so-called tension field, a simplification is introduced in order to reduce computational efforts. However, this model was restricted to linear elastic material models in previous works. Aiming to develop an implicit isogeometric wrinkling model for large strain and hyperelastic material applications, a modified deformation gradient has been assumed, which can be used for any strain energy density formulation. The model is an extension of a previously published model for linear elastic material behaviour and is generalised to other types of discretisation as well. The extension for hyperelastic materials requires the derivative of the material tensor, which can be computed numerically or derived analytically. The presented model relies on a combination of dynamic relaxation and a Newton–Raphson solver, because of divergence in early Newton–Raphson iterations as a result of a changing tension field, which is not included in the stress tensor variation. Using four benchmarks, the model performance is evaluated. Convergence with the expected order for Newton–Raphson iterations has been observed, provided a fixed tension field. The model accurately approximates the mean surface of a wrinkled membrane with a reduced number of degrees of freedom in comparison to a shell solution.

This paper proposes a novel approach for solving nonlinear partial differential equations (PDEs) with a quantum computer, the trainable embedding quantum physics informed neural network (TE-QPINN). We combine quantum machine learning (QML) with physics informed neural networks (PINNs) in a hybrid approach. By leveraging the advantages of classical and quantum computers, we can create algorithms that have a potential to be run on noisy intermediate-scale quantum devices (NISQ). We use feedforward neural networks (FNN) as problem-agnostic embedding functions, giving the used quantum circuit greater expressibility than previously introduced embedding. This expressibility allows us to solve a wide range of problems without using a problem specific ansatz. Additionally, we introduce a hybrid backpropagation algorithm that allows efficient updates of the used weights and biases in the FNN embedding functions. In this paper we showcase the capabilities of TE-QPINNs of a wide range of problems, including the two-dimensional Poisson, Burgers and Navier-Stokes equations. In direct comparison with classical PINNs, this approach showed an ability to achieve superior results while using the same number of parameters, highlighting their potential for more efficient optimization in high-dimensional parameter spaces, which could be transformative for future applications.

The increasing reliance on 3D scanning and meshless methods highlights the need for algorithms optimized for point-cloud geometry representations in CAE simulations. While voxel-based binning methods are simple, they often compromise geometry and topology, particularly with coarse voxelizations. We propose an algorithm based on a Series of Local Triangulations (SOLT) as an intermediate representation for point clouds, enabling efficient upsampling and downsampling. This robust and straightforward approach preserves the integrity of point clouds, ensuring resampling without feature loss or topological distortions. The proposed techniques integrate seamlessly into existing engineering workflows, avoiding complex optimization or machine learning methods while delivering reliable, high-quality results for a large number of examples. Resampled point clouds produced by our method can be directly used for solving PDEs or as input for surface reconstruction algorithms. We demonstrate the effectiveness of this approach with examples from mechanically sampled point clouds and real-world 3D scans.

Benchmarking quantum computers helps to quantify them and bringing the technology to the market. Various application-level metrics exist to benchmark a quantum device at an application level. This paper presents a revised holistic scoring method called the Quantum Application Score (QuAS) incorporating strong points of previous metrics, such as QPack and the Q-score. We discuss how to integrate both and thereby obtain an application-level metric that better quantifies the practical utility of quantum computers. We evaluate the new metric on different hardware platforms such as D- Wave and IBM as well as quantum simulators of Quantum Inspire and Rigetti.

Part-scale thermal process simulations play an important role in improving the part quality of the Laser Powder Bed Fusion (LPBF) process. The semi-analytical simulation method relies on the superposition of analytical fields to represent laser-induced heat sources in a semi-infinite space and a complementary temperature field to enforce boundary conditions. So far, boundary conditions have been imposed by analytical image fields for straight boundaries and numerically for non-straight boundaries. The latter requires considerable refinement on the spatial discretization, at least near the boundaries, and compromises the computational efficiency of the simulations. In this paper, we derive a closed-form solution for the image fields that can accurately enforce the boundary conditions for non-straight boundaries. A geometrically complex part boundary is represented by B-splines, and with the aid of an offset method and reparameterization, the positions of the image sources are determined. The image field's closed-form expression is then found using the boundary's local curvature calculated from the local tangent lines. Numerical examples on different levels of complexity revealed that the net heat lost along an adiabatic boundary vanishes when the novel image source solutions are used, and the thermal evolution of complex parts can be accurately predicted with high computational efficiency. Simulations involving multiple lasers can also be performed with no extra computational cost.

This paper explores the potential application of quantum and hybrid quantum–classical neural networks in power flow analysis. Experiments are conducted using two datasets based on 4-bus and 33-bus test systems. A systematic performance comparison is also conducted among quantum, hybrid quantum–classical, and classical neural networks. The comparison is based on (i) generalization ability, (ii) robustness, (iii) training dataset size needed, (iv) training error, and (v) training process stability. The results show that the developed hybrid quantum–classical neural network outperforms both quantum and classical neural networks, and hence can improve deep learning-based power flow analysis in the noisy-intermediate-scale quantum (NISQ) and fault-tolerant quantum (FTQ) era.

We propose a method for optimizing the geometry of a freeform lens to redirect the light emitted from an extended source into a desired irradiance distribution. We utilize a gradient-based optimization approach with MITSUBA 3, an algorithmic differentiable non-sequential ray tracer that allows us to obtain the gradients of the freeform surface parameters with respect to the produced irradiance distribution. To prevent the optimizer from getting trapped in local minima, we gradually increase the number of degrees of freedom of the surface by using Truncated Hierarchical B-splines (THB-splines) during optimization. The refinement locations are determined by analyzing the gradients of the surface vertices. We first design a freeform using a collimated beam (zero-etendue source) for a complex target distribution to demonstrate the method’s effectiveness. Then, we demonstrate the ability of this approach to create a freeform that can project the light of an extended Lambertian source into a prescribed target distribution.

Noisy intermediate-scale quantum computers (NISQ) are computing hardware in their childhood, but they are showing high promise and growing quickly. They are based on so-called qubits, which are the quantum equivalents of bits. Any given qubit state results in a given probability of observing a value of zero or one in the readout process. One of the main concerns for NISQ machines is the inherent noisiness of qubits, i.e., the observable frequencies of zeros and ones do not correspond to the theoretically expected probability, as the qubit states are subject to random disturbances over time and with each additional algorithmic operation applied to them. Models to describe the influence of this noise exist.In this study, we conduct extensive experiments on quantum noise. Based on our data, we show that existing noise models lack important aspects. Specifically, they fail to properly capture the aggregation of noise effects over time (or over an algorithm's runtime), and they are underdispersed. With underdispersion, we refer to the fact that observable frequencies scatter much more between repeated experiments than what the standard assumptions of the binomial distribution would allow for. Based on these shortcomings, we develop an extended noise model for the probability distribution of observable frequencies as a function of the number of gate operations. The model roots back to a known continuous random walk on the (Bloch) sphere, where the angular diffusion coefficient can be used to characterize the standard noisiness of gate operations. Here, we superimpose a second random walk at the scale of multiple readouts to account for overdispersion. Further, our model has known, explicit components for noise during state preparation and measurement. The interaction of these two random walks predicts theoretical, runtime-dependent bounds for probabilities. Overall, it is a three-parameter distributional model that fits the data much better than the corresponding one-scale model (without overdispersion), and we demonstrate the better fit and the plausibility of the predicted bounds via Bayesian data-model analysis.

In this paper we present a scalable algorithm for fault-tolerant quantum computers for solving the transport equation in two and three spatial dimensions for variable grid sizes and discrete velocities, where the object walls are aligned with the Cartesian grid, the relative difference of velocities in each dimension is bounded by 1 and the total simulated time is dependent on the discrete velocities chosen. We provide detailed descriptions and complexity analyses of all steps of our quantum transport method (QTM) and present numerical results for 2D flows generated in Qiskit as a proof of concept. Our QTM is based on a novel streaming approach which leads to a reduction in the amount of CNOT gates required in comparison to state-of-the-art quantum streaming methods. As a second highlight of this paper we present a novel object encoding method, that reduces the complexity of the amount of CNOT gates required to encode walls, which now becomes independent of the size of the wall. Finally we present a novel quantum encoding of the particles' discrete velocities that enables a linear speed-up in the costs of reflecting the velocity of a particle, which now becomes independent of the amount of velocities encoded. Our main contribution consists of a detailed description of a fail-safe implementation of a quantum algorithm for the reflection step of the transport equation that can be readily implemented on a physical quantum computer. This fail-safe implementation allows for a variety of initial conditions and particle velocities and leads to physically correct particle flow behavior around the walls, edges and corners of obstacles. Combining these results we present a novel and fail-safe quantum algorithm for the transport equation that can be used for a multitude of flow configurations and leads to physically correct behavior. We finally show that our approach only requires O(n_wn_g²+dn_t^vn_vmax²) CNOT gates, which is quadratic in the amount of qubits necessary to encode the grid and the amount of qubits necessary to encode the discrete velocities in a single spatial dimension. This complexity result makes our approach superior to state-of-the-art approaches known in the literature.

Parallel computing is omnipresent in today's scientific computer landscape, starting at multicore processors in desktop computers up to massively parallel clusters. While domain decomposition methods have a long tradition in computational mechanics to decompose spatial problems into multiple subproblems that can be solved in parallel, advancing solution schemes for dynamics or quasi-statics are inherently serial processes. For quasi-static simulations, however, there is no accumulating ‘time’ discretization error, hence an alternative approach is required. In this paper, we present an Adaptive Parallel Arc-Length Method (APALM). By using a domain parametrization of the arc-length instead of time, the multi-level error for the arc-length parametrization is formed by the load parameter and the solution norm. Given coarse approximations of arc-length intervals, finer corrections enable the parallelization of the presented method. This results in an arc-length method that is parallel within a branch and inherently adaptive. This concept is easily extended for bifurcation problems. The performance of the method is demonstrated using isogeometric Kirchhoff-Love shells on problems with snap-through and pitch-fork instabilities and applied to the problem of a snapping meta-material. These results show that parallel corrections are performed in a fraction of the time of the serial initialization, achievable on desktop scale.

As with many tasks in engineering, structural design frequently involves navigating complex and computationally expensive problems. A prime example is the weight optimization of laminated composite materials, which to this day remains a formidable task, due to an exponentially large configuration space and non-linear constraints. The rapidly developing field of quantum computation may offer novel approaches for addressing these intricate problems. However, before applying any quantum algorithm to a given problem, it must be translated into a form that is compatible with the underlying operations on a quantum computer. Our work specifically targets stacking sequence retrieval with lamination parameters, which is typically the second phase in a common bi-level optimization procedure for minimizing the weight of composite structures. To adapt stacking sequence retrieval for quantum computational methods, we map the possible stacking sequences onto a quantum state space. We further derive a linear operator, the Hamiltonian, within this state space that encapsulates the loss function inherent to the stacking sequence retrieval problem. Additionally, we demonstrate the incorporation of manufacturing constraints on stacking sequences as penalty terms in the Hamiltonian. This quantum representation is suitable for a variety of classical and quantum algorithms for finding the ground state of a quantum Hamiltonian. For a practical demonstration, we performed numerical state-vector simulations of two variational quantum algorithms and additionally chose a classical tensor network algorithm, the DMRG algorithm, to numerically validate our approach. For the DMRG algorithm, we derived a matrix product operator representation of the loss function Hamiltonian and the penalty terms. Although this work primarily concentrates on quantum computation, the application of tensor network algorithms presents a novel quantum-inspired approach for stacking sequence retrieval.

We investigate the performance of different annealers for power flow analysis using adiabatic computing. The annealers include D-Wave's simulated annealer Neal, D-Wave's quantum-classical hybrid annealer, D-Wave's Advantages system (QA), Fujitsu's classical simulated annealer, and Fujitsu's digital annealer V3 (DA). We implement Quadratic Unconstrained Binary Optimization (QUBO) and Ising model formulations, with the latter offering finer control over complex voltage adjustments. Different test systems are experimented with to systematically evaluate the annealers. The evaluation is based on the accuracy, the annealer's capability to handle the decision variables, and the computational time needed. QA and DA show superior performance over classical annealers for our application. DA effectively manages larger test systems, whereas QA encounters difficulties embedding the problem graph onto the hardware graph because of the limited qubit connectivity. This constraint confines QA to the 14-bus system with the QUBO formulation and the 4-bus system with the Ising model formulation. The best performance is associated with different annealers across different test systems, which suggests that adjusting the threshold can improve precision if the compiler and annealer are capable of handling the number of variables involved.

Mesh adaptivity is a technique to provide detail in numerical solutions without the need to refine the mesh over the whole domain. Mesh adaptivity in isogeometric analysis can be driven by Truncated Hierarchical B-splines (THB-splines) which add degrees of freedom locally based on finer B-spline bases. Labeling of elements for refinement is typically done using residual-based error estimators. In this paper, an adaptive meshing workflow for isogeometric Kirchhoff–Love shell analysis is developed. This framework includes THB-splines, mesh admissibility for combined refinement and coarsening and the Dual-Weighted Residual (DWR) method for computing element-wise error contributions. The DWR can be used in several structural analysis problems, allowing the user to specify a goal quantity of interest which is used to mark elements and refine the mesh. This goal functional can involve, for example, displacements, stresses, eigenfrequencies etc. The proposed framework is evaluated through a set of different benchmark problems, including modal analysis, buckling analysis and non-linear snap-through and bifurcation problems, showing high accuracy of the DWR estimator and efficient allocation of degrees of freedom for advanced shell computations.

Isogeometric analysis has brought a paradigm shift in integrating computational simulations with geometric designs across engineering disciplines. This technique necessitates analysis-suitable parameterization of physical domains to fully harness the synergy between Computer-Aided Design and Computer-Aided Engineering analyses. Existing methods often fix boundary parameters, leading to challenges in elongated geometries such as fluid channels and tubular reactors. This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations. We employ a sophisticated Schwarz–Christoffel mapping technique, which is instrumental in computing boundary correspondences. A refined boundary curve reparameterization process complements this. Our dual-strategy approach maintains the geometric exactness and continuity of input physical domains, overcoming limitations often encountered with the existing reparameterization techniques. By employing our proposed boundary parameter matching method, we show that even a simple linear interpolation approach can effectively construct a satisfactory analysis-suitable parameterization. Our methodology offers significant improvements over traditional practices, enabling the generation of analysis-suitable and geometrically precise models, which is crucial for ensuring accurate simulation results. Numerical experiments show the capacity of the proposed method to enhance the quality and reliability of isogeometric analysis workflows.

In recent years, quantum Boltzmann methods have gained more and more interest as they might provide a viable path toward solving fluid dynamics problems on quantum computers once this emerging compute technology has matured and fault-tolerant many-qubit systems become available. The major challenge in developing a start-to-end quantum algorithm for the Boltzmann equation consists in encoding relevant data efficiently in quantum bits (qubits) and formulating the streaming, collision and reflection steps as one comprehensive unitary operation. The current literature on quantum Boltzmann methods mostly proposes data encodings and quantum primitives for individual phases of the pipeline, assuming that they can be combined to a full algorithm. In this paper, we disprove this assumption by showing that for encodings commonly discussed in the literature, either the collision or the streaming step cannot be unitary. Building on this landmark result, we propose a novel encoding in which the number of qubits used to encode the velocity depends on the number of time steps one wishes to simulate, with the upper bound depending on the total number of grid points. In light of the non-unitarity result established for existing encodings, our encoding method is to the best of our knowledge the only one currently known that can be used for a start-to-end quantum Boltzmann solver where both the collision and the streaming step are implemented as a unitary operation.

The past years have seen a surge in quantum algorithms for computational fluid dynamics (CFD). These algorithms have in common that whilst promising a speed-up in the performance of the algorithm, no specific method of measurement has been suggested. This means that while the algorithms presented in the literature may be promising methods for creating the quantum state that represents the final flow field, an efficient measurement strategy is not available. This paper marks the first quantum method proposed to efficiently calculate quantities of interest (QoIs) from a state vector representing the flow field. In particular, we propose a method to calculate the force acting on an object immersed in the fluid using a quantum version of the momentum exchange method (MEM) that is commonly used in lattice Boltzmann methods to determine the drag and lift coefficients. In order to achieve this we furthermore give a scheme that implements bounce back boundary conditions on a quantum computer, as those are the boundary conditions the momentum exchange method is designed for.

Power flow (PF) analysis is a foundational computational method to study the flow of power in an electrical network. This analysis involves solving a set of non-linear and non-convex differential-algebraic equations. State-of-the-art solvers for PF analysis, therefore, face challenges with scalability and convergence, specifically for large-scale and/or ill-conditioned cases characterized by high penetration of renewable energy sources, among others. The adiabatic quantum computing paradigm has been proven to efficiently find solutions for combinatorial problems in the noisy intermediate-scale quantum (NISQ) era, and it can potentially address the limitations posed by state-of-the-art PF solvers. For the first time, we propose a novel adiabatic quantum computing approach for efficient PF analysis. Our key contributions are (i) a combinatorial PF algorithm and a modified version that aligns with the principles of PF analysis, termed the adiabatic quantum PF algorithm (AQPF), both of which use Quadratic Unconstrained Binary Optimization (QUBO) and Ising model formulations; (ii) a scalability study of the AQPF algorithm; and (iii) an extension of the AQPF algorithm to handle larger problem sizes using a partitioned approach. Numerical experiments are conducted using different test system sizes on D-Wave’s Advantage™ quantum annealer, Fujitsu’s digital annealer V3, D-Wave’s quantum-classical hybrid annealer, and two simulated annealers running on classical computer hardware. The reported results demonstrate the effectiveness and high accuracy of the proposed AQPF algorithm and its potential to speed up the PF analysis process while handling ill-conditioned cases using quantum and quantum-inspired algorithms.