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IEEE Asia-Pacific Conference on Computer Science and Data Engineering (CSDE)

Solving the Bounded-Depth Steiner Tree Problem using an Adiabatic Quantum Computer

Kai Liu, Michael J. Dinneen

Published Dec 1, 2019• 6 citations
Publisher

Abstract

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.

DOI
10.1109/csde48274.2019.9162395

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The Steiner Tree Problem (STP) is a well-known NP-hard combinatorial optimization problem, which has wide applications in network design, integrated circuit layout, bioinformatics, and other fields. However, traditional algorithms often struggle to balance efficiency and solution quality when dealing with large-scale STP instances. In this paper, we propose a new quantum annealing-based algorithm for solving the STP: we first model the STP into a quadratic unconstrained binary optimization (QUBO) form suitable for quantum annealing, then design a corresponding encoding strategy, and finally verify the algorithm through experimental tests. The results show that our quantum annealing-based method can obtain high-quality solutions with relatively low computational overhead for moderate-scale STP instances, providing a new feasible path for handling this intractable combinatorial optimization problem.
Quantum computers are early-stage technology, but are progressing rapidly. Here, we compare metaheuristics with a quantum annealer (QA), a quantum computer tailored to solve combinatorial optimisation problems formulated as a Quadratic Unconstrained Binary Optimisation (QUBO) problem. Formulating non-trivial optimisation problems as QUBOs is a challenge, so it is to ensure that the direct comparison of classical and quantum search algorithms is conducted fairly. This paper illustrates the use of a streamlined methodology to derive a QUBO for a classical optimisation problem – the Steiner tree problem – and provides comprehensive experiments comparing D-WAVE pure QA and hybrid QA with classical methods including Simulated Annealing and Genetic Algorithms. We find that the pure QA suffers from considerable qubit noise but the hybrid QA is on a par with traditional methods.
We review existing quantum computational methods for solving the Hamiltonian cycle problem in different computational frameworks such as quantum circuits, quantum walks and adiabatic quantum computation. Then we present a QUBO (quadratic unconstrained binary optimization) formulation, which is suitable for the adiabatic quantum computation for a D-Wave architecture. Further, we derive a physical Hamiltonian from the QUBO formulation and discuss its adequateness in the adiabatic framework. Finally, we discuss the complexity of running the Hamiltonian cycle QUBO on a D-Wave quantum computer, and compare it with existing quantum computational methods.
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 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.