Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named quantum-inspired classical algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.

翻译：自开创性的HHL量子线性系统求解算法及随后Kerenidis与Prakash提出的量子推荐系统算法以来，为重要机器学习任务实现可证明的指数级量子加速已成为核心研究目标。这些算法最初被认为是实现指数级加速的有力候选方案，但排除类似经典改进的下界证明始终缺失。在Tang的突破性工作中，她证明了经典下界研究缺乏进展具有充分理由。具体而言，她提出了量子推荐系统算法的经典对应版本，将量子优势削弱至多项式级别。该方法具有普适性，被命名为量子启发经典算法。此后，几乎所有最初的指数级量子机器学习加速均通过新型量子启发经典算法被降级为多项式加速。基于当前研究现状，我们尚不清楚能否在任何自然机器学习任务中期待指数级量子加速。本工作首次针对线性系统求解这一基础问题，在输入矩阵良态且行列稀疏的条件下，给出了量子算法与量子启发经典算法之间可证明的指数级分离。