We present a quantum algorithmic routine that extends the realm of Grover-based heuristics for tackling combinatorial optimization problems with arbitrary efficiently computable objective and constraint functions. Building on previously developed quantum methods that were primarily restricted to linear constraints, we generalize the approach to encompass a broader class of problems in discrete domains. To evaluate the potential of our algorithm, we assume the existence of sufficiently advanced logical quantum hardware. With this assumption, we demonstrate that our method has the potential to outperform state-of-the-art classical solvers and heuristics in terms of both runtime scaling and solution quality. The same may be true for more realistic implementations, as the logical quantum algorithm can achieve runtime savings of up to $10^2-10^3$.
We present a quantum algorithmic routine that extends the realm of Grover-based heuristics for tackling combinatorial optimization problems with arbitrary efficiently computable objective and constraint functions. Building on previously developed quantum methods that were primarily restricted to linear constraints, we generalize the approach to encompass a broader class of problems in discrete domains. To evaluate the potential of our algorithm, we assume the existence of sufficiently advanced logical quantum hardware. With this assumption, we demonstrate that our method has the potential to outperform state-of-the-art classical solvers and heuristics in terms of both runtime scaling and solution quality. The same may be true for more realistic implementations, as the logical quantum algorithm can achieve runtime savings of up to 102−103.
Quantum computing is a rapidly growing field, promising computational advancements with significant impact on real-world and industrial applications. These areas include cryptography, financial systems, material and drug discovery. They also cover combinatorial optimization, which is the focus of this thesis. For combinatorial optimization problems, researchers have spent several decades extending the capabilities of quantum algorithms for both near-term and far-term quantum hardware. Because current devices comprise too few qubits (quantum bits) and noisy gates (qubit operations prone to errors), potential quantum advantage must be assessed using theoretical and numerical frameworks. Despite the existence of many quantum algorithms, such as Grover’s algorithm and Quantum Branch-and-Bound (QB&B), which provide asymptotic speedups, we lack evidence of a practical quantum speedup on realistic problem instances. Recent benchmarking efforts show that several far-term algorithms with asymptotic advantage still cannot achieve practical quantum advantage under realistic conditions. This thesis comprises three advancements. First, it develops a new type of state preparation tailored to constraint optimization problems. This enables a Grover-based heuristic we call Constraint-oriented biased quantum search (CBQS). Second, it introduces benchmarking methods that compare the quantum search with state-of-the-art classical algorithms on large instances. These methods indicate that the proposed quantum algorithm can achieve a practical quantum advantage. For some cases, logical gate times of just 10−6s with about 19,000 logical qubits are enough to match classical runtime. Third, it presents a quantum circuit backend (software that processes quantum circuits) designed for speed at the cost of suboptimal circuit design. This backend outperforms a wide range of existing software packages in both speed and memory consumption, demonstrated by Quantum Fourier Transformation (QFT) circuits of up to 2000 qubits. Overall, this work points the way toward full-stack quantum advantage while underscoring the need for further research on topics such as quantum error correction.
The semiconductor supply chain gives rise to many complex, large and heavily constrained combinatorial optimization problems. We derive simpler problems from these complex formulations and analyze quantum algorithms for their solution. We study two main regimes: variational algorithms derived from the Quantum Approximate Optimization Algorithm (QAOA) and quantum search algorithms based on Grover’s search algorithm and Amplitude Amplification. In the first case, the main challenge lies in integrating problem constraints into the QAOA framework, which was originally devised for unconstrained problems such as MaxCut. The canonical method of adding penalty terms to the objective leads to an undesirable inflation of the energy spectrum. Instead, we examine alternative problem encodings as well as constraint-preserving mixing operators, which are designed to limit the quantum algorithm to the subspace of feasible states. We make explicit suggestions for a wide selection of problems. In the second case, we operate under the premise that any quantum search protocol for combinatorial optimization must use tailored initial states– otherwise the square root speedup of quantum search is not enough to offset the exponential size of the search space in a combinatorial optimization problem. Consequently, we examine state preparation routines for a wide range or problems. The main devices lie in preparing superpositions of only feasible states and superpositions which are centered around a reference solution. We present a benchmarking technique, in an exact and an approximate version, which allows us to generate performance metrics for our algorithms based on quantum search. We use the benchmarking tools to carry out numerical studies for a selection of problems. We conclude that our variational methods tailored to constrained problems while technically sound– lead to deep quantum circuits which are not suitable for the NISQ machines which QAOA was originally developed for. Our conclusion for optimization algorithms based on quantum search is more optimisitic. Their compatability with classical optimization principles, such as neighborhood search, gives rise to promising further research directions. Advanced benchmarking techniques allow us to test these algorithms independently of quantum hardware developments.
This study explores quantum computing for combinatorial optimization, focusing on the Traveling Salesman Problem (TSP). Using Qiskit, the Quantum Approximate Optimization Algorithm (QAOA) is implemented on simulators and quantum hardware, and compared with classical optimization using Gurobi. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with quadratic penalties enforcing degree and subtour constraints. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound methods achieve better solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, performance degrades with problem size due to sensitivity to parameter tuning, penalty scaling, and hardware noise. Among the tested classical optimizers, COBYLA provides the best balance of efficiency and noise tolerance. Overall, this work establishes a reproducible quantum-classical workflow, provides performance baselines, and identifies critical factors such as encoding design, optimizer choice, and error mitigation for improving near-term quantum heuristic performance.
To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing is realized, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem and for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of depth-1 QAOA, a parameterized quantum algorithm, by deriving a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.
Constrained combinatorial optimization (CCO) problems are prevalent across various fields and represent key challenges in computational science and engineering. Although numerous classical and quantum algorithms have been proposed to tackle these problems, substantial limitations still persist. Classical algorithms exhibit exponential computational complexity growth with scale and persistent vulnerability to local minima traps. Quantum computing, while offering theoretical advantages through global superposition, faces practical barriers such as short decoherence times and current hardware limitations. To address these challenges, we propose a quantum-inspired fast algorithm for solving CCO problems. Our approach enhances global search capability with superposition encoding and avoids constraint-induced local minima via a project-feedback strategy. Particularly, our method aligns with mature electronics manufacturing and demonstrates a proof-of-concept implementation in classical systems, indicating high-efficiency potential for solving constrained optimization problems.