Computational methods are rapidly emerging as an essential tool for understanding and solving complex engineering problems, which complement the traditional tools of experimentation and theory. When considered in a discrete computational setting, many engineering problems can be reduced to a graph coloring problem. Examples range from systems design, airline scheduling, image segmentation to pattern recognition, where energy cost functions with discrete variables are extremized. However, using discrete variables over continuous variables introduces some complications when defining differential quantities, such as gradients and Hessians involved in scientific computations within solid and fluid mechanics. Consequently, graph techniques are under-utilized in this important domain. However, we have recently witnessed great developments in quantum computing where physical devices can solve discrete optimization problems faster than most well-known classical algorithms. This warrants further investigation into the re-formulation of scientific computation problems into graph-theoretic problems, thus enabling rapid engineering simulations in a soon-to-be quantum computing world. The computational techniques developed in this thesis allow the representation of surface scalars, such as perimeter and area, using discrete variables in a graph. Results from integral geometry, specifically Cauchy-Crofton relations, are used to estimate these scalars via submodular functions. With this framework, several quantities important to engineering applications can be represented in graph-based algorithms. These include the surface energy of cracks for fracture prediction, grain boundary energy to model microstructure evolution, and surface area estimates (of grains and fibers) for generating conformal meshes. Combinatorial optimization problems for these applications are presented first. The last two chapters describe two new graph coloring algorithms implemented on a physical quantum computing device: the D-wave quantum annealer. The first algorithm describes a functional minimization approach to solve differential equations. The second algorithm describes a realization of the Boltzmann machine learning algorithm on a quantum annealer. The latter allows generative and discriminative learning of data, which has vast applications in many fields. Theoretical aspects and the implementation of these problems are outlined with a focus on engineering applications.
Computational methods are rapidly emerging as an essential tool for understanding and solving complex engineering problems, which complement the traditional tools of experimentation and theory. When considered in a discrete computational setting, many engineering problems can be reduced to a graph coloring problem. Examples range from systems design, airline scheduling, image segmentation to pattern recognition, where energy cost functions with discrete variables are extremized. However, using discrete variables over continuous variables introduces some complications when defining differential quantities, such as gradients and Hessians involved in scientific computations within solid and fluid mechanics. Consequently, graph techniques are under-utilized in this important domain. However, we have recently witnessed great developments in quantum computing where physical devices can solve discrete optimization problems faster than most well-known classical algorithms. This warrants further investigation into the re-formulation of scientific computation problems into graph-theoretic problems, thus enabling rapid engineering simulations in a soon-to-be quantum computing world. The computational techniques developed in this thesis allow the representation of surface scalars, such as perimeter and area, using discrete variables in a graph. Results from integral geometry, specifically Cauchy-Crofton relations, are used to estimate these scalars via submodular functions. With this framework, several quantities important to engineering applications can be represented in graph-based algorithms. These include the surface energy of cracks for fracture prediction, grain boundary energy to model microstructure evolution, and surface area estimates (of grains and fibers) for generating conformal meshes. Combinatorial optimization problems for these applications are presented first. The last two chapters describe two new graph coloring algorithms implemented on a physical quantum computing device: the D-wave quantum annealer. The first algorithm describes a functional minimization approach to solve differential equations. The second algorithm describes a realization of the Boltzmann machine learning algorithm on a quantum annealer. The latter allows generative and discriminative learning of data, which has vast applications in many fields. Theoretical aspects and the implementation of these problems are outlined with a focus on engineering applications.
Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below. Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing. A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach. We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy. We consider the simulation of quantum mechanical systems on a quantum computer. We show a novel divide and conquer approach for Hamiltonian simulation. Using the Hamiltonian structure, we can obtain faster simulation algorithms. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under mild assumptions. We turn to combinatorial optimization problems. An important open question is whether quantum computers provide advantages for the approximation of classically hard combinatorial problems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Optimization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem, and derive analytic performance bounds for the lowest circuit-depth realization, for both general and special classes of graphs. Along the way, we develop a general procedure for analyzing the performance of QAOA for other problems, and show an example demonstrating the difficulty of obtaining similar results for greater depth. We show a generalization of QAOA and its application to wider classes of combinatorial optimization problems, in particular, problems with feasibility constraints. We introduce the Quantum Alternating Operator Ansatz, which utilizes more general unitary operators than the original QAOA proposal. Our framework facilitates low-resource implementations for many applications which may be particularly suitable for early quantum computers. We specify design criteria, and develop a set of results and tools for mapping diverse problems to explicit quantum circuits. We derive constructions for several important prototypical problems including Maximum Independent Set, Graph Coloring, and the Traveling Salesman problem, and show appealing resource cost estimates for their implementations.
Quantum computing holds the potential to efficiently solve certain problems believed to be intractable for classical computers. Realizing this potential, however, requires substantial physical resources, including large qubit counts, sufficient connectivity, and long coherence times. As current quantum devices are still far from this regime, the efficient compilation of quantum algorithms into hardware-compatible instructions remains a central challenge in quantum computing, especially for near-term applications. A representative example of such applications is combinatorial optimization, where exponentially large search spaces motivate quantum algorithms that leverage interference to approximate solutions more efficiently than classical heuristics, even at modest system sizes. Two prominent paradigms in this context are quantum annealing and the Quantum Approximate Optimization Algorithm (QAOA): Quantum annealing realizes optimization via continuous-time quantum dynamics implemented on analog hardware, while QAOA is a variational, gate-based approach that builds parameterized quantum circuits from problem-specific Hamiltonians. Both paradigms require compilation strategies tailored to the constraints of their respective hardware architectures. Most existing compilation strategies are problem-agnostic and therefore broadly applicable, yet often suboptimal for specific tasks. This thesis uses the graph coloring problem as a case study to investigate a problem-driven approach to quantum compilation, where structural features of the problem class are exploited to design tailored compilation schemes that offer improved efficiency and scalability for the targeted application. To address hardware connectivity constraints, the presented approaches adopt the parity framework, which encodes long-range interactions into single-qubit terms. This thesis develops two novel problem-driven compilation methods for the graph coloring problem, the Bandwidth-Optimized LHZ layout and the 2D-Twine circuit. The former leverages bandwidth minimization to optimize parity encodings for analog quantum annealers, while the latter constructs QAOA unitaries for gate-based optimization using parallelized Parity Twine circuits. Both approaches are benchmarked against general-purpose methods and demonstrate substantial improvements in qubit usage or circuit complexity. These results illustrate the advantages of incorporating problem structure into the compilation process and provide a proof-of-concept for the effectiveness of problem-driven compilation strategies.
Challenging combinatorial optimization problems are ubiquitous in science and engineering. Several quantum methods for optimization have recently been developed, in different settings including both exact and approximate solvers. Addressing this field of research, this manuscript has three distinct purposes. First, we present an intuitive method for synthesizing and analyzing discrete ( i . e . , integer-based) optimization problems, wherein the problem and corresponding algorithmic primitives are expressed using a discrete quantum intermediate representation (DQIR) that is encoding-independent. This compact representation often allows for more efficient problem compilation, automated analyses of different encoding choices, easier interpretability, more complex runtime procedures, and richer programmability, as compared to previous approaches, which we demonstrate with a number of examples. Second, we perform numerical studies comparing several qubit encodings; the results exhibit a number of preliminary trends that help guide the choice of encoding for a particular set of hardware and a particular problem and algorithm. Our study includes problems related to graph coloring, the traveling salesperson problem, factory/machine scheduling, financial portfolio rebalancing, and integer linear programming. Third, we design low-depth graph-derived partial mixers (GDPMs) up to 16-level quantum variables, demonstrating that compact (binary) encodings are more amenable to QAOA than previously understood. We expect this toolkit of programming abstractions and low-level building blocks to aid in designing quantum algorithms for discrete combinatorial problems.
Quantum computing is a rapidly developing field that has seen a huge amount of interest in the last couple of decades due to its promise of revolutionizing several domains of business and science. It presents a new way of doing computations by making use of fundamental properties of quantum mechanics such as superposition and entanglement. Optimization, on the other hand, is a field that is omnipresent in the industry and where small improvements can have a significant impact. This thesis aims to tackle optimization problems using quantum algorithms.NP-hard optimization problems are not believed to be exactly solvable through general polynomial time algorithms. Variational quantum algorithms (VQAs) to address such combinatorial problems have been of great interest recently. Such algorithms are heuristic and aim to obtain an approximate solution. The hardware, however, is still in its infancy and the current Noisy Intermediate Scale Quantum (NISQ) computers are not able to optimize industrially relevant problems. Moreover, the storage of qubits and introduction of entanglement require extreme physical conditions.An issue with contemporary quantum optimization algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) is that they scale linearly with problem size. To tackle this issue, we present the LogQ encoding, using which we can design quantum variational algorithms that scale logarithmically with problem size - opening an avenue for treating optimization problems of unprecedented scale on gate-based quantum computers. We show how this algorithm can be applied to several combinatorial optimization problems such as Maximum Cut, Minimum Partition, Maximum Clique and Maximum Weighted Independent Set (MWIS). Subsequently, these algorithms are tested on a quantum simulator with graph sizes of over a hundred nodes and on a real quantum computer up to graph sizes of 256. To our knowledge, these constitute the largest realistic combinatorial optimization problems ever run on a NISQ device, overcoming previous problem sizes by almost tenfold.Next, we apply the LogQ encoding to two use-cases for large companies such as TotalEnergies. Fleet conversion is the process of transitioning a fleet of vehicles to more sustainable and environmentally friendly alternatives. It is modeled as a column generation scheme with the MWIS problem as the sub-problem or worker problem. We use the LogQ method to solve the MWIS Workers and demonstrate how quantum and classical solvers can be used together in a hybrid manner to approach an industrial-sized use-case. Mesh segmentation refers to the process of dividing a complex mesh (composed of vertices, edges, and faces) into meaningful and semantically coherent parts or regions. Mesh segmentation plays an important part in computer modeling, which is extensively used in the core domains of TotalEnergies' activities such as Earth imaging, physical modeling for reservoirs, and others. We define the problem as a graph optimization problem and use the LogQ encoding to solve it.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.
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.