Recently, several quantum machine learning algorithms have been proposed that may offer quantum speed-ups over their classical counterparts. Most of these algorithms are either heuristic or assume that data can be accessed quantum-mechanically, making it unclear whether a quantum advantage can be proven without resorting to strong assumptions. Here we construct a classification problem with which we can rigorously show that heuristic quantum kernel methods can provide an end-to-end quantum speed-up with only classical access to data. To prove the quantum speed-up, we construct a family of datasets and show that no classical learner can classify the data inverse-polynomially better than random guessing, assuming the widely believed hardness of the discrete logarithm problem. Furthermore, we construct a family of parameterized unitary circuits, which can be efficiently implemented on a fault-tolerant quantum computer, and use them to map the data samples to a quantum feature space and estimate the kernel entries. The resulting quantum classifier achieves high accuracy and is robust against additive errors in the kernel entries that arise from finite sampling statistics.

最近，已经提出了几种量子机器学习算法，这些算法可能会比经典算法提供量子加速。这些算法中的大多数要么是启发式的，要么假设可以通过量子力学访问数据，因此不清楚是否可以在不诉诸强假设的情况下证明量子优势。在这里，我们构建了一个分类问题，我们可以用它严格证明启发式量子核方法可以提供端到端的量子加速，只有经典的数据访问。为了证明量子加速，我们构建了一系列数据集，并表明假设离散对数问题的难度被广泛认为，没有经典学习器可以比随机猜测更好地对数据进行逆多项式分类。此外，我们构建了一系列参数化的单位电路，它可以在容错的量子计算机上有效地实现，并使用它们将数据样本映射到量子特征空间并估计内核条目。由此产生的量子分类器实现了高精度，并且对内核条目中由有限采样统计产生的附加误差具有鲁棒性。