In recent years, strong expectations have been raised for the possible power of quantum computing for solving difficult optimization problems, based on theoretical, asymptotic worst-case bounds. Can we expect this to have consequences for Linear and Integer Programming when solving instances of practically relevant size, a fundamental goal of Mathematical Programming, Operations Research and Algorithm Engineering? Answering this question faces a crucial impediment: The lack of sufficiently large quantum platforms prevents performing real-world tests for comparison with classical methods. In this paper, we present a quantum analog for classical runtime analysis when solving real-world instances of important optimization problems. To this end, we measure the expected practical performance of quantum computers by analyzing the expected gate complexity of a quantum algorithm. The lack of practical quantum platforms for experimental comparison is addressed by hybrid benchmarking, in which the algorithm is performed on a classical system, logging the expected cost of the various subroutines that are employed by the quantum versions. In particular, we provide an analysis of quantum methods for Linear Programming, for which recent work has provided asymptotic speedup through quantum subroutines for the Simplex method. We show that a practical quantum advantage for realistic problem sizes would require quantum gate operation times that are considerably below current physical limitations.

翻译：近年来，基于理论渐近最坏情况界，量子计算在解决困难优化问题方面可能具有强大能力的预期被广泛提出。当求解实际相关规模的问题实例时——这是数学规划、运筹学和算法工程的基本目标——我们能否期望这一能力对线性规划和整数规划产生实际影响？回答这一问题的关键障碍在于：缺乏足够大规模的量子平台，无法进行真实世界测试以与经典方法进行对比。在本文中，我们提出了一种量子模拟，用于经典运行时分析以求解实际相关规模的重要优化问题实例。为此，我们通过分析量子算法的预期门复杂度来度量量子计算机的预期实际性能。针对缺乏实际量子平台进行实验比较的问题，我们采用混合基准测试方法：在经典系统上执行算法，同时记录量子版本所采用的各个子程序的预期成本。具体而言，我们提供了线性规划量子方法的分析——近期工作已通过单纯形法的量子子程序实现了渐近加速。研究表明，对于实际相关的问题规模，实现实际量子优势所需的量子门操作时间需显著低于当前物理限制。