We experimentally evaluate the practical state-of-the-art in graph bipartization (Odd Cycle Transversal (OCT)), motivated by the need for good algorithms for embedding problems into near-term quantum computing hardware. We assemble a preprocessing suite of fast input reduction routines from the OCT and Vertex Cover (VC) literature and compare algorithm implementations using Quadratic Unconstrained Binary Optimization problems from the quantum literature. We also generate a corpus of frustrated cluster loop graphs, which have previously been used to benchmark quantum annealing hardware. The diversity of these graphs leads to harder OCT instances than in existing benchmarks. In addition to combinatorial branching algorithms for solving OCT directly, we study various reformulations into other NP-hard problems such as VC and Integer Linear Programming (ILP), enabling the use of solvers such as CPLEX. We find that for heuristic solutions with time constraints under a second, iterative compression routines jump-started with a heuristic solution perform best, after which point using a highly tuned solver like CPLEX is worthwhile. Results on exact solvers are split between using ILP formulations on CPLEX and solving VC formulations with a branch-and-reduce solver. We extend our results with a large corpus of synthetic graphs, establishing robustness and potential to generalize to other domain data. In total, over 8,000 graph instances are evaluated, compared to the previous canonical corpus of 100 graphs. Finally, we provide all code and data in an open source suite, including a Python API for accessing reduction routines and branching algorithms, along with scripts for fully replicating our results.
We experimentally evaluate the practical state-of-the-art in graph bipartization (Odd Cycle Transversal (OCT)), motivated by the need for good algorithms for embedding problems into near-term quantum computing hardware. We assemble a preprocessing suite of fast input reduction routines from the OCT and Vertex Cover (VC) literature and compare algorithm implementations using Quadratic Unconstrained Binary Optimization problems from the quantum literature. We also generate a corpus of frustrated cluster loop graphs, which have previously been used to benchmark quantum annealing hardware. The diversity of these graphs leads to harder OCT instances than in existing benchmarks. In addition to combinatorial branching algorithms for solving OCT directly, we study various reformulations into other NP-hard problems such as VC and Integer Linear Programming (ILP), enabling the use of solvers such as CPLEX. We find that for heuristic solutions with time constraints under a second, iterative compression routines jump-started with a heuristic solution perform best, after which point using a highly tuned solver like CPLEX is worthwhile. Results on exact solvers are split between using ILP formulations on CPLEX and solving VC formulations with a branch-and-reduce solver. We extend our results with a large corpus of synthetic graphs, establishing robustness and potential to generalize to other domain data. In total, over 8,000 graph instances are evaluated, compared to the previous canonical corpus of 100 graphs. Finally, we provide all code and data in an open source suite, including a Python API for accessing reduction routines and branching algorithms, along with scripts for fully replicating our results.
We experimentally evaluate the practical state-of-the-art in graph bipartization (Odd Cycle Transversal), motivated by recent advances in near-term quantum computing hardware and the related embedding problems. We assemble a preprocessing suite of fast input reduction routines from the Odd Cycle Transversal (OCT) and Vertex Cover (VC) literature, allowing algorithm implementations to be compared using Quadratic Unconstrained Binary Optimization problems from the quantum literature. These problems represent a wide array of graph properties, leading to harder OCT instances than in previous benchmarks.
In addition to combinatorial branching algorithms for solving OCT directly, we study various reformulations into other NP-hard problems such as VC and Integer Linear Programming (ILP), enabling the use of solvers such as CPLEX. We find that for heuristic solutions with time constraints under a second, iterative compression routines jump-started with a heuristic solution perform best, after which point using a highly tuned solver like CPLEX is worthwhile. Results on exact solvers are split between using ILP formulations on CPLEX and solving VC formulations with a branch-and-reduce solver. We compare our results to those on a corpus of synthetic graphs, establishing robustness and potential to generalize to other domain data.
Finally, we provide all code and data in an open source suite, including a Python API for accessing reduction routines and branching algorithms, along with scripts for fully replicating our results.
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.
Benchmarking Quantum Process Units (QPU) at an application level usually requires considering the whole programming stack of the quantum computer. One critical task is the minor-embedding (resp. transpilation) step, which involves space-time overheads for annealing-based (resp. gate-based) quantum computers. This paper establishes a new protocol to generate graph instances with their associated near-optimal minor-embedding mappings to D-Wave Quantum Annealers (QA). This set of favorable mappings is used to generate a wide diversity of optimization problem instances. We use this method to benchmark QA on large instances of unconstrained and constrained optimization problems and compare the performance of the QPU with efficient classical solvers. The benchmark aims to evaluate and quantify the key characteristics of instances that could benefit from the use of a quantum computer. In this context, existing QA seem best suited for unconstrained problems on instances with densities less than 10%. For constrained problems, the penalty terms used to encode the hard constraints restrict the performance of QA and suggest that these QPU will be less efficient on these problems of comparable size.
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 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.
Combinatorial optimisation is a main topic for real-world applications, where fast solutions are required. Optimisation algorithms are incredibly powerful, but still face bottlenecks for certain problems. In recent years, quantum algorithms have been proposed as a potential candidate for solving such problems. The question that naturally arises is whether they could provide meaningful speed-ups over classical approaches. The study of asymptotic complexity has become central in the development of quantum algorithms. Nevertheless, having a promising asymptotic improvement does not directly translate into good practical performance. This is already well known for classical algorithms, which can be executed and evaluated directly. When it comes to quantum algorithms, the situation is more nuanced, as solving practically relevant instances would require a fully fault-tolerant quantum computer. In this thesis, we address the challenge of assessing the practical performance of a quantum algorithm without access to quantum hardware. Our strategy is to evaluate how these algorithms would perform under idealised fault-tolerant assumptions and to identify parameter regimes in which quantum approaches could potentially become useful. Our goal is to guide algorithmic development long before implementation becomes possible. We first give a general overview of the quantum (sub)routines that serve as building blocks for more advanced algorithms, as well as of the three hybrid benchmarking techniques used throughout this thesis. Then, we elaborate on each method: we start with the algorithmic descriptions, proceed to the methodology, and conclude with their hybrid evaluation on practically relevant datasets. Within the hybrid benchmarking method 1 (gate count analysis), we analytically derive the minimum number of gates that a quantum version of the classical simplex algorithm would require to solve realistic instances. By executing classical counterparts of the quantum subroutines, we infer the execution speed required per gate for the quantum simplex to outperform its classical counterpart. Within the hybrid benchmarking method 2 (query count analysis), we benchmark functional quantum linear solvers by analytically deriving a precise number of query calls made to the same oracles as a function of problem parameters. This provides insight into which algorithm might be more suitable in a given parameter regime. It allows for a direct comparison only between different quantum algorithms and is independent of classical runtime measurements. Within the hybrid benchmarking method 3 (cycle count analysis), we develop a runtime estimation framework tailored to the quantum tree generator–based search for solving 0-1-knapsack problems. We evaluate its performance on large scale instances beyond the reach of general simulation. We compare it with state-of-the-art classical linear programming solvers in terms of processor cycle counts. Hybrid benchmarking provides a fair, systematic benchmarking methodology, ensuring reproducible and relevant comparisons not only between classical and quantum approaches, but also among different quantum algorithms.