Quantum physics has the potential to revolutionize computer science by building what is called a quantum computer. Quantum computers can perform tasks that are impossible with Turing-machine based classical computers (e.g., generating random sequences of bits) and can outperform classical computers (to different degrees) in a variety of classical computation problems. Several different models of quantum computing exist. Based on the Adiabatic Theorem in physics, Adiabatic Quantum Computing is a novel computing framework that is drastically different from the more traditional quantum logic gate model. Traditionally, quantum computing has only been tested on devices with a very small number of qubits, severely limiting their uses in practice. In recent years however, advances in engineering and physics have allowed the construction for (relatively) large machines called quantum annealers that do what is known as a specific type of adiabatic quantum computation called quantum annealing. Solving classical computation problems with the quantum annealing framework is not an easy task and there are many obstacles to overcome if we want to achieve any kind of speedups over classical algorithms. In this thesis, we present quantum annealing solutions to several classical problems including the Hamiltonian Cycle Problem, Mixed Dominating Set Problem, Graph Isomorphism Problem, Densest k-Subgraph Problem and the Maximum Weighted Independent Set Problem. Experimental results of these quantum solutions on DWave quantum annealers are also reported. We also look at some of the bottlenecks in the quantum annealing framework and propose several novel approach to mitigate their effects on the efficiency of the quantum solutions.
Quantum physics has the potential to revolutionize computer science by building what is called a quantum computer. Quantum computers can perform tasks that are impossible with Turing-machine based classical computers (e.g., generating random sequences of bits) and can outperform classical computers (to different degrees) in a variety of classical computation problems. Several different models of quantum computing exist. Based on the Adiabatic Theorem in physics, Adiabatic Quantum Computing is a novel computing framework that is drastically different from the more traditional quantum logic gate model. Traditionally, quantum computing has only been tested on devices with a very small number of qubits, severely limiting their uses in practice. In recent years however, advances in engineering and physics have allowed the construction for (relatively) large machines called quantum annealers that do what is known as a specific type of adiabatic quantum computation called quantum annealing. Solving classical computation problems with the quantum annealing framework is not an easy task and there are many obstacles to overcome if we want to achieve any kind of speedups over classical algorithms. In this thesis, we present quantum annealing solutions to several classical problems including the Hamiltonian Cycle Problem, Mixed Dominating Set Problem, Graph Isomorphism Problem, Densest k-Subgraph Problem and the Maximum Weighted Independent Set Problem. Experimental results of these quantum solutions on DWave quantum annealers are also reported. We also look at some of the bottlenecks in the quantum annealing framework and propose several novel approach to mitigate their effects on the efficiency of the quantum solutions.
Quantum computation provides exponential speedup for solving certain mathematical problems against classical computers. Motivated by current rapid experimental progress on quantum computing devices, various models of quantum computation have been investigated to show quantum computational supremacy. At a commercial side, quantum annealing machine realizes the quantum Ising model with a transverse field and heuristically solves combinatorial optimization problems. The computational power of this machine is closely related to adiabatic quantum computation (AQC) with a restricted type of Hamiltonians, namely stoquastic Hamiltonians, and has been thought to be relatively less powerful compared to universal quantum computers. Little is known about computational quantum speedup nor advantage in AQC with stoquastic Hamiltonians. Here we characterize computational capability of AQC with stoquastic Hamiltonians, which we call stoqAQC. We construct a concrete stoqAQC model, whose lowest energy gap is lower bounded polynomially, and hence the final state can be obtained in polynomial time. Then we show that it can simulate universal quantum computation if adaptive single-qubit measurements in non-standard bases are allowed on the final state. Even if the measurements are restricted to non-adaptive measurements to respect the robustness of AQC, the proposed model exhibits quantum computational supremacy; classical simulation is impossible under complexity theoretical conjectures. Moreover, it is found that such a stoqAQC model can simulate Shor's algorithm and solve the factoring problem in polynomial time. We also propose how to overcome the measurement imperfections via quantum error correction within the stoqAQC model and also an experimentally feasible verification scheme to test whether or not stoqAQC is done faithfully.
Abstract Several fundamental questions demand space exploration, posing tremendous computational tasks to address modelling and data processing. Quantum computing has been proposed to address deep space-related problems from quantum gravity, nuclear processes and many-body field theory simulation to artificial intelligence and related space mission planning. Adiabatic quantum computers have been proposed for solving optimization problems, such as space mission scheduling. The problem of assigning jobs to resources in a scientific mission, while satisfying certain constraints, can be tackled with the help of D-Wave’s quantum annealer, which exploits the adiabatic theorem to find the minimum energy solutions of a given system, encoded as the Hamiltonian of a quadratic unconstrained binary optimization (QUBO) problem. In its simplest em-bodiement, the scheduling can be implemented as a graph coloring (GC) problem, where the goal is to assign different colors (out of a given set) to connected nodes of an input graph, while respecting given constraints. The feasibility of our methods and the performance of quantum annealers are studied on randomly generated GC instances. A second framework studied is the job shop scheduling problem (JSP) that encodes more real-like scheduling problems of interest to agencies and industries. By implementing the problems with a QUBO formulation, we are able to find the optimal solution to three JSP instances present in the literature. The biggest one is solved with a quantum computation approach here for the first time.
Quantum computing has emerged as a promising computational paradigm with the potential to solve problems that are intractable for classical computers. This paper provides an overview of the current state of quantum computing, including its historical development, hardware advancements, applications, and challenges. Special attention is given to combinatorial optimization, which is considered one of the most promising application domains. The limitations of current quantum systems and future research directions are also discussed. This paper presents a comparative study between a conventional D-Wave-based simulated annealing approach and a proposed hybrid optimization model for solving largescale combinatorial optimization problems. The study focuses on benchmark problems, including AES, TSP gr24, and a real-world load balancing problem, all formulated as QUBO models. The D-Wave approach employs the dwave-neal simulated annealing engine with various parameter settings, while the proposed model introduces a decomposition-based framework combined with adaptive parameter control and iterative refinement. The experimental results demonstrate that the proposed model consistently outperforms the conventional approach in terms of solution quality, achieving significantly lower energy values across all problem instances. In particular, for the TSP gr24 problem, the proposed model achieves near-optimal solutions with substantially reduced computation time. For large-scale problems such as AES and load balancing, the proposed model maintains scalability and robustness, whereas the conventional approach suffers from increased computational cost and performance degradation. Although the proposed method introduces additional computational steps, its hybrid structure effectively mitigates limitations related to local minima, parameter sensitivity, and scalability. These findings indicate that integrating classical optimization techniques with quantum-inspired methods is essential for practical applications. The proposed model provides a promising framework for solving complex real-world optimization problems and highlights the importance of hybrid approaches in advancing quantum annealing technologies.
We provide efficient quadratic unconstrained binary optimization (QUBO) formulations for the Dominating Set and Edge Cover combinatorial problems suitable for adiabatic quantum computers, which are viewed as a real-world enhanced model of simulated annealing (e.g. a type of genetic algorithm with quantum tunneling). The number of qubits (dimension of QUBO matrices) required to solve these set cover problems are O(n + n lg n) and O(m + n lg n) respectively, where n is the number of vertices and m is the number of edges. We also extend our formulations for the Minimum Vertex-Weighted Dominating Set problem and the Minimum Edge-Weighted Edge Cover problem. Experimental results for the Dominating Set and Edge Cover problems using a D-Wave Systems quantum computer with 1098 active qubit-coupled processors are also provided for a selection of known common graphs.
Quantum computing requires qubits, which obey the laws of quantum mechanics and therefore behave very dierently from classical bits. This dierence can allow quantum computers to achieve massive levels of parallelism impossible on a classical computer. As a result, certain problems that are in NP but highly parallelizable can be solved by quantum computers in polynomial time. Quantum computers have the capacity to drastically change the computing world, but they are complex and dicult to implement; to study them, we often look to simulators that can bypass many of the roadblocks inhibiting our path to a true quantum computer. A simulated quantum annealer takes advantage of the adiabatic theorem to solve problems by transitioning from a simple to complex quantum state while maintaining minimal energy. We simulate these quantum states, energies, and transitions on classical machines by mathematical abstractions for proof of concept and to circumvent the complications of physical implementation. To do this, we solve 3SAT on a classical machine simulating how it would be solved by quantum annealing.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.