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arXiv 2202.01636quant-phcs.DS
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关于图优化问题的常数时间量子退火和保证近似

On constant-time quantum annealing and guaranteed approximations for graph optimization problems

Arthur Braida, Simon Martiel, Ioan Todinca

摘要:

量子退火(QA)是一种计算框架,其中量子系统的连续演化用于在非结构化搜索空间上找到目标函数的全局最小值。它可以被看作是一个通用的元启发式优化问题,包括NP难的,如果我们允许一个指数级大的运行时间。虽然QA被广泛研究,从启发式的角度来看,很少有人知道的理论保证在多项式时间内获得的解决方案的质量。在本文中,我们使用的技术借用理论物理学,Lieb-Robinson(LR)界,并开发新的工具证明,短,恒定时间量子退火保证常数因子近似比的一些优化问题时,限制到有界度图。非正式地,在有界度图的LR界允许我们检索(放松)局部参数,通过它可以推导出的近似比通过研究有界半径的子图。我们说明了我们的工具上的问题最大割和最大独立集的三次图,提供明确的近似比和运行时需要获得它们。我们的结果是相似的味道,在不同的,但相关的QAOA(量子优化算法)框架中获得的著名的。最后,我们讨论了进一步改进的理论和实验参数。

英文摘要:

Quantum Annealing (QA) is a computational framework where a quantum system's continuous evolution is used to find the global minimum of an objective function over an unstructured search space. It can be seen as a general metaheuristic for optimization problems, including NP-hard ones if we allow an exponentially large running time. While QA is widely studied from a heuristic point of view, little is known about theoretical guarantees on the quality of the solutions obtained in polynomial time. In this paper we use a technique borrowed from theoretical physics, the Lieb-Robinson (LR) bound, and develop new tools proving that short, constant time quantum annealing guarantees constant factor approximations ratios for some optimization problems when restricted to bounded degree graphs. Informally, on bounded degree graphs the LR bound allows us to retrieve a (relaxed) locality argument, through which the approximation ratio can be deduced by studying subgraphs of bounded radius. We illustrate our tools on problems MaxCut and Maximum Independent Set for cubic graphs, providing explicit approximation ratios and the runtimes needed to obtain them. Our results are of similar flavor to the well-known ones obtained in the different but related QAOA (quantum optimization algorithms) framework. Eventually, we discuss theoretical and experimental arguments for further improvements.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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