Combinatorial optimization - a field of research addressing problems that feature strongly in a wealth of scientific and industrial contexts - has been identified as one of the core potential fields of applicability of quantum computers. It is still unclear, however, to what extent quantum algorithms can actually outperform classical algorithms for this type of problems. In this work, by resorting to computational learning theory and cryptographic notions, we prove that quantum computers feature an in-principle super-polynomial advantage over classical computers in approximating solutions to combinatorial optimization problems. Specifically, building on seminal work by Kearns and Valiant and introducing a new reduction, we identify special types of problems that are hard for classical computers to approximate up to polynomial factors. At the same time, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The core of the quantum advantage discovered in this work is ultimately borrowed from Shor's quantum algorithm for factoring. Concretely, we prove a super-polynomial advantage for approximating special instances of the so-called integer programming problem. In doing so, we provide an explicit end-to-end construction for advantage bearing instances. This result shows that quantum devices have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms. Our results also give clear guidance on how to construct such advantage-bearing problem instances.

翻译：组合优化——一个研究在科学和工业领域中广泛出现的问题的领域——已被视为量子计算机的核心潜在应用领域之一。然而，对于此类问题，量子算法在多大程度上能实际超越经典算法仍不明确。在本工作中，通过借助计算学习理论和密码学概念，我们证明了量子计算机在原则上具有超越经典计算机的多项式级优势，能够逼近组合优化问题的解。具体而言，基于Kearns和Valiant的开创性工作并引入一种新的归约，我们识别出对经典计算机而言难以在多项式因子内逼近的特殊问题类型。同时，我们给出一种量子算法，能够高效地在多项式因子内逼近最优解。本工作发现的量子优势的核心最终源自Shor的量子因式分解算法。具体地，我们证明了在逼近所谓整数规划问题的特殊实例时存在超越多项式的优势。在此过程中，我们为具有优势的实例提供了显式的端到端构造。这一结果表明，量子设备在原则上具备超越经典高效算法能力来逼近组合优化问题的潜力。我们的结果还为如何构造此类优势问题实例提供了明确指导。