Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that plays an important role in networking, computational biology, and biochemistry. We show an explicit construction of Ising Hamiltonians whose ground states encode the solution of SCP instances. We numerically simulate the time-dependent Schrödinger equation in order to test the performance of quantum annealing for random instances and compare with that of simulated annealing. We also discuss explicit embedding strategies for realizing our Hamiltonian construction on the D-wave type restricted Ising Hamiltonian based on Chimera graphs. Our embedding on the Chimera graph preserves the structure of the original SCP instance and in particular, the embedding for general complete bipartite graphs and logical disjunctions may be of broader use than that the specific problem we deal with.
Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that plays an important role in networking, computational biology, and biochemistry. We show an explicit construction of Ising Hamiltonians whose ground states encode the solution of SCP instances. We numerically simulate the time-dependent Schrödinger equation in order to test the performance of quantum annealing for random instances and compare with that of simulated annealing. We also discuss explicit embedding strategies for realizing our Hamiltonian construction on the D-wave type restricted Ising Hamiltonian based on Chimera graphs. Our embedding on the Chimera graph preserves the structure of the original SCP instance and in particular, the embedding for general complete bipartite graphs and logical disjunctions may be of broader use than that the specific problem we deal with.
Quantum annealing (QA) uses the principles of quantum mechanics for solving unconstrained optimization problems. Since its initial proposal, there has been much interest in the search for practical problems for which QA has advantage over classical algorithms. Although it is believed that quantum computers cannot in general solve NP-complete problems efficiently, there has been evidence suggesting that effects such as quantum tunneling can bring quantum speedup over classical computation for some problems in NP. In this dissertation, we consider two problems that have both theoretical and practical importance. One is the Set Cover with Pairs problem. The other is the Factorization problem. The Set Cover with Pairs (SCP) problem is a generalization of the Set Cover problem. It requires each element in a set to be covered by at least two objects instead of one as in the Set Cover problem. The SCP problem is believed to be not only NP-hard, but also hard to approximate. In this dissertation, we explicitly construct the Ising Hamiltonians whose ground states encode the solutions of SCP instances. The resulting Ising Hamiltonian has the appealing property that the control precision required for encoding an SCP instance scales only linearly with the number of objects in the covering set. The Integer Factorization problem is of great interest in the quantum community because it is difficult to solve using classical methods and its importance in cryptography. In this dissertation, we propose a general framework for solving factorization problems using quantum annealing, by mapping the framework to an Ising Hamiltonian. We develop several methods to consider the specific requirements such as control precision, spin-spin connection in the current Ising machines. We test the methods on the latest D-Wave 2000Q machine.
Quantum annealing (QA) uses the principles of quantum mechanics for solving unconstrained optimization problems. Since its initial proposal, there has been much interest in the search for practical problems for which QA has advantage over classical algorithms. Although it is believed that quantum computers cannot in general solve NP-complete problems effciently, there has been evidence suggesting that effects such as quantum tunneling can bring quantum speedup over classical computation for some problems in NP. In this dissertation, we consider two problems that have both theoretical and practical importance. One is the Set Cover with Pairs problem. The other is the Factorization problem. The Set Cover with Pairs (SCP) problem is a generalization of the Set Cover problem. It requires each element in a set to be covered by at least two objects instead of one as in the Set Cover problem. The SCP problem is believed to be not only NP-hard, but also hard to approximate. In this dissertation, we explicitly construct the Ising Hamiltonians whose ground states encode the solutions of SCP instances. The resulting Ising Hamiltonian has the appealing property that the control precision required for encoding an SCP instance scales only linearly with the number of objects in the covering set. The Integer Factorization problem is of great interest in the quantum community because it is diffcult to solve using classical methods and its importance in cryptography. In this dissertation, we propose a general framework for solving factorization problems using quantum annealing, by mapping the framework to an Ising Hamiltonian. We develop several methods to consider the specifc requirements such as control precision, spin-spin connection in the current Ising machines. We test the methods on the latest D-Wave 2000Q machine.
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.
The recent availability of the first commercial quantum computers has provided a promising tool to tackle NP hard problems which can only be solved heuristically with present techniques. However, it is unclear if the current state of quantum computing already provides a quantum advantage over the current state of the art in classical computing. This article assesses the performance of the D-Wave 2X quantum annealer on two NP hard graph problems, in particular clique finding and graph partitioning. For this, we provide formulations as Qubo and Ising Hamiltonians suitable for the quantum annealer and compare a variety of quantum solvers (Sapi, QBSolv, QSage provided by D-Wave Sys, Inc.) to current classical algorithms (METIS, Simulated Annealing, third-party clique finding and graph splitting heuristics) on certain test sets of graphs. We demonstrate that for small graph instances, classical methods still outperform the quantum annealer in terms of computing time, even though the quality of the best solutions obtained is comparable. Nevertheless, due to the limited problem size which can be embedded on the D-Wave 2X chip, the aforementioned finding applies to most of problems of general nature solvable on the quantum annealer. For instances specifically designed to fit the D-Wave 2X architecture, we observe substantial speed-ups in computing time over classical approaches.
Quantum annealing (QA) can be used to quickly obtain near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum odd cycle transversals (OCTs). One of the formulations presented is exact and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller. Summary of Contribution: Our work combines classical and quantum computing for operations research by showing that integer linear programming can be successfully used as a preprocessing step for adiabatic quantum optimization. We use it to determine how a quadratic unconstrained binary optimization problem can be solved by a quantum annealer in which the qubits are coupled as in a Chimera graph, such as in the quantum annealers currently produced by D-Wave Systems. The paper also provides a timely introduction to adiabatic quantum computing and related work on minor embeddings.
Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine, D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. In addition, we also evaluate the performance of a heuristic by Selby. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue. In particular, we examine how reliably the D-Wave computer can deliver true optimum solutions and present some surprising results.