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Australasian Computer Science Week Multiconference (ACSW)

Formulating graph covering problems for adiabatic quantum computers

Michael J. Dinneen, Richard Hua

Published Dec 20, 2016• 15 citations
Publisher

Abstract

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.

DOI
10.1145/3014812.3014830

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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.
We propose a quadratic unconstrained binary optimization (QUBO) formulation for the Bounded-Depth Steiner Tree problem suitable for adiabatic quantum computers. The numbers of qubits (dimension of QUBO matrices) required by our formulation is O(|V|3), where |V| represents the number of vertices. A D-Wave 2X computer consisting of 1098 active qubits is evaluated for a selection of known common graphs. Experimental results are shown and discussed. Our results are applicable to the Bounded-Depth Minimum Spanning Tree problem, as a special case.
This article addresses the question of implementing a maximum flow algorithm on directed graphs in a formulation suitable for a quantum annealing computer. Three distinct approaches are presented. In all three cases, the flow problem is formulated as a quadratic unconstrained binary optimization (QUBO) problem amenable to quantum annealing. The first implementation augments a graph with integral edge capacities into a multigraph with unit-capacity edges and encodes the fundamental objective and constraints of the maximum flow problem using a number of qubits equal to the total capacity of the graph ∑ici\sum _i{c_i}∑i​ci​. The second implementation, which encodes flows through edges using a binary representation, reduces the required number of qubits to O(∣E∣log⁡Cmax⁡)\mathcal {O}(|E| \log C_{\max })O(∣E∣logCmax​), where ∣E∣|E|∣E∣ and Cmax⁡C_{\max }Cmax​ denote the number of edges and maximum edge capacity of the graph, respectively. The third implementation adapts the dual minimum cut formulation and encodes the problem instance using ∣V∣|V|∣V∣ qubits, where ∣V∣|V|∣V∣ is the number of vertices in the graph. Scaling factors for penalty terms and coupling matrix construction times are made explicit in this article.
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.
The problem of finding the maximum weighted cycle in a directed graph maps to optimization problems in NP, implying that approaches in classical computing are inefficient. Here, Quantum computing might be a promising alternative. Many current approaches on the quantum computer are based on a Quadratic Unconstrained Binary Optimization (QUBO) problem formulation. This paper develops four different QUBO approaches to this problem. The first two take the starting vertex and the number of vertices used in the cycle as given, while the latter two loosen the second assumption of knowing the size of the cycle. A QUBO formulation is derived for each approach. Further, the number of binary variables required to encode the maximum weighted cycle problem with one or both assumptions for the respective approach is made explicit. The problem is motivated by finding the maximum weighted debt cycle in a debt graph. This paper compares classical computing versus currently avaiable (hybrid) quantum computing approaches for various debt graphs. For the classical part, it investigated the Depth-First-Search (DFS) method and Simulated Annealing. For the (hybrid) quantum approaches, we utilized direct embedding on the quantum annealer and two types of quantum hybrid solvers. Simulated Annealing and the usage of the hybrid CQM (Constrained Quadratic Model) machine functioned promising. The DFS method, direct QPU, and hybrid BQM (Binary Quadratic Model), on the other hand, performed less, due to memory issues, surpassing the limit of decision variables, and finding the right penalty values, respectively.
Abstract It has been demonstrated that Binary Integer Programming (BIP) formulations can be reformulated to a corresponding quadratic unconstrained binary optimization (QUBO) problem. The reformulation allows for BIPs to run as QUBOs on adiabatic quantum annealing hardware. Current BIP to QUBO reformulation techniques propagate dense QUBO structures that are not ideal for current hardware where quantum bits, and the connections between them, are scarce. We propose a methodology for BIP to QUBO reformulation that results in a sparse QUBO that, when used to embed set-partitioning problems onto a D-Wave Pegasus topology, requires 80% fewer qubits and is embedded is 41 times faster. The same methodology for set-covering problems requires 77% fewer qubits and is embedded 26 times faster. We discuss the limitations of the technique and avenues for future extensions.
Quantum annealing algorithms were introduced to solve combinatorial optimization problems by taking advantage of quantum fluctuations to escape local minima in complex energy landscapes typical of NP - hard problems. In this work, we propose using quantum annealing for the theory of cuts, a field of paramount importance in theoretical computer science. We have proposed a method to formulate the Minimum Multicut Problem into the QUBO representation, and the technical difficulties faced when embedding and submitting a problem to the quantum annealer processor. We show two constructions of the quadratic unconstrained binary optimization functions for the Minimum Multicut Problem and we review several tradeoffs between the two mappings and provide numerical scaling analysis results from several classical approaches. Furthermore, we discuss some of the expected challenges and tradeoffs in the implementation of our mapping in the current generation of D-Wave machines.