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IEEE Congress on Evolutionary Computation (CEC)

First Results Solving Arbitrarily Structured Maximum Independent Set Problems Using Quantum Annealing

Sheir Yarkoni, Aske Plaat, Thomas Bäck

Published Jul 1, 2018• 29 citations
Publisher

Abstract

Commercial quantum processing units (QPUs) such as those made by D-Wave Systems are being increasingly used for solving complex combinatorial optimization problems. In this paper, we review a canonical NP-hard problem, the Maximum Independent Set (MIS) problem. We show how to map MIS problems to quadratic unconstrained binary optimization (QUBO) problems, and use a D-Wave 2000Q QPU to solve them. We compare the results from the D-Wave system to classical algorithms such as simulated thermal annealing and the graphical networks package NetworkX. To our knowledge, these are the first results of experiments involving arbitrarily-structured MIS inputs using a D-Wave QPU. We find that the QPU can be used as a heuristic optimizer for randomly generated inputs, but due to physical control errors, can be outperformed by simulated thermal annealing.

DOI
10.1109/cec.2018.8477865

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ACM International Conference on Computing Frontiers (CF)•May 15, 2017•31 citations
Commercial quantum processing units (QPUs) such as those made by D-Wave Systems are being increasingly used for solving complex combinatorial optimization problems. In this paper, we review a canonical NP-hard problem, the Maximum Independent Set (MIS) problem. We show how to map MIS problems to quadratic unconstrained binary optimization (QUBO) problems, and use a D-Wave 2000Q QPU to solve them. We compare the results from the D-Wave system to classical algorithms such as simulated thermal annealing and the graphical networks package NetworkX. To our knowledge, these are the first results of experiments involving arbitrarily-structured MIS inputs using a D-Wave QPU. We find that the QPU can be used as a heuristic optimizer for randomly generated inputs, but due to physical control errors, can be outperformed by simulated thermal annealing.
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.
Quantum annealers can be used to solve many (possibly NP-hard) combinatorial optimization problems, by formulating them as quadratic unconstrained binary optimization (QUBO) problems or, equivalently, using the Ising formulation. In this paper we analyse the ability of quantum D-Wave annealers to find the maximum clique on a graph, expressed as a QUBO problem. Due to the embedding limit of 164 nodes imposed by the anneler, we conducted a study on graph decomposition to enable instance embedding. We thus propose a decomposition algorithm for the complementary maximum independent set problem, and a graph generation algorithm to control the number of nodes, the number of cliques, the density, the connectivity indices and the ratio of the solution size to the number of other nodes. We then statistically analysed how these variables affect the quality of the solutions found by the quantum annealer. The results of our investigation include recommendations on ratio and density limits not to be exceeded, as well as a series of precautions and a priori analyses to be carried out in order to maximise the probability of obtaining a solution close to the optimum.
We present experiments in solving combinatorial optimization and constraint satisfaction problems by means of Quantum Annealing. We describe how to model classic constraint problems such as N-queens and Magic Square as well as hard combinatorial problems such as the Costas Array Problem or the Quadratic Assignment Problem in terms of QUBO (Quadratic Unconstrained Binary Optimization). QUBO is the input language of quantum computers based on quantum annealing such as the D-Wave systems and of the ”quantum-inspired” but classical devices such as Fujitsu’s Digital Annealing Unit or Hitachi’s CMOS Annealing Machine. We present preliminary results for solving these combinatorial optimization and constraint satisfaction problems by implementation on the D-Wave quantum computer.
The Maximum Independent Set (MIS) problem is a fundamental combinatorial optimization task that can be naturally mapped onto the Ising Hamiltonian of neutral atom quantum processors. Given its connection to NP-hard problems and real-world applications, there has been significant experimental interest in exploring quantum advantage for MIS. Pioneering experiments on King's Lattice graphs suggested a quadratic speed-up over simulated annealing, but recent benchmarks using state-of-the-art methods found no clear advantage, likely due to the structured nature of the tested instances. In this work, we generate hard instances of unit-disk graphs by leveraging complexity theory results and varying key hardness parameters such as density and treewidth. For a fixed graph size, we show that increasing these parameters can lead to prohibitive classical runtime increases of several orders of magnitude. We then compare classical and quantum approaches on small instances and find that, at this scale, quantum solutions are slower than classical ones for finding exact solutions. Based on extended classical benchmarks at larger problem sizes, we estimate that scaling up to a thousand atoms with a 1 kHz repetition rate is a necessary step toward demonstrating a computational advantage with quantum methods.
This paper assesses the performance of the D-Wave 2X (DW) quantum annealer for finding a maximum clique in a graph, one of the most fundamental and important NP-hard problems. Because the size of the largest graphs DW can directly solve is quite small (usually around 45 vertices), we also consider decomposition algorithms intended for larger graphs and analyze their performance. For smaller graphs that fit DW, we provide formulations of the maximum clique problem as a quadratic unconstrained binary optimization (QUBO) problem, which is one of the two input types (together with the Ising model) acceptable by the machine, and compare several quantum implementations to current classical algorithms such as simulated annealing, Gurobi, and third-party clique finding heuristics. We further estimate the contributions of the quantum phase of the quantum annealer and the classical post-processing phase typically used to enhance each solution returned by DW. We demonstrate that on random graphs that fit DW, no quantum speedup can be observed compared with the classical algorithms. On the other hand, for instances specifically designed to fit well the DW qubit interconnection network, we observe substantial speed-ups in computing time over classical approaches.