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 a super-polynomial advantage over classical computers in approximating solutions to combinatorial optimization problems. Specifically, building on seminal work by Kearns and Valiant, we have identified special types of problems that are hard for classical computers to approximate. Despite this, we give a quantum algorithm such that an optimal solution can be efficiently approximated by quantum computers. The advantage for special instances of the so-called integer programming problem is shown to be super-polynomial. This result shows that quantum devices have 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——研究在众多科学和工业情境中具有重要地位的各类问题的领域——已被确定为量子计算机潜在的核心应用领域之一。然而，对于这类问题，量子算法在多大程度上能够实际超越经典算法，目前仍不明确。在本研究中，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逼近组合优化问题解的能力上拥有超越经典计算机的超多项式优势。具体而言，基于Kearns与Valiant的开创性工作，我们识别出某些特殊类型的问题，经典计算机难以对其进行逼近。尽管如此，我们提出了一种量子算法，使得量子计算机能够高效地逼近最优解。对于所谓的整数规划问题的特殊实例，这种优势被证明是超多项式的。这一结果表明，量子设备具有逼近组合优化解的能力，远超经典高效算法的范畴。我们的研究结果还为如何构造此类具有优势的问题实例提供了明确指导。