6 research outputs found
Finding Maximum Cliques on a Quantum Annealer
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
Comparing Three Generations of D-Wave Quantum Annealers for Minor Embedded Combinatorial Optimization Problems
Quantum annealing is a novel type of analog computation that aims to use
quantum mechanical fluctuations to search for optimal solutions of Ising
problems. Quantum annealing in the Transverse Ising model, implemented on
D-Wave QPUs, are available as cloud computing resources. In this article we
report concise benchmarks across three generations of D-Wave quantum annealers,
consisting of four different devices, for the NP-Hard combinatorial
optimization problems unweighted maximum clique and unweighted maximum cut on
random graphs. The Ising, or equivalently QUBO, formulation of these problems
do not require auxiliary variables for order reduction, and their overall
structure and weights are not highly complex, which makes these problems simple
test cases to understand the sampling capability of current D-Wave quantum
annealers. All-to-all minor embeddings of size , with relatively uniform
chain lengths, are used for a direct comparison across the Chimera, Pegasus,
and Zephyr device topologies. A grid search over annealing times and the minor
embedding chain strengths is performed in order to determine the level of
reasonable performance for each device and problem type. Experiment metrics
that are reported are approximation ratios for non-broken chain samples and
chain break proportions. How fairly the quantum annealers sample optimal
maximum cliques, for instances which contain multiple maximum cliques, is also
quantified using entropy of the measured ground state distributions. The newest
generation of quantum annealing hardware, which has a Zephyr hardware
connectivity, performed the best overall with respect to approximation ratios
and chain break frequencies
Um Algoritmo HÃbrido para o Problema da Clique Máxima / A Hybrid Algorithm for the Maximum Clique Problem
Este artigo apresenta um algoritmo que realiza a hibridização entre a meta-heurÃstica Simulated Anneling e uma Lista Tabu para resolver o problema da clique máxima. O algoritmo foi avaliado mediante os resultados obtidos para o banco de instâncias do Centro de matemática discreta e ciência da computação teórica (DIMACS). O algoritmo consegue processar um total de 83% das instâncias em menos de 2 minutos e obtém o valor ótimo para 74,6%. Os resultados mostraram-se promissores em contraste com as observações realizadas na literatura. Em comparação com o resultado recente publicado, a média da solução do algoritmo aqui apresentado foi estritamente melhor em 52,11% e empataram em 21,1% das instâncias