It has been hypothesized that quantum computers may lend themselves well to applications in machine learning. In the present work, we analyze function classes defined via quantum kernels. Quantum computers offer the possibility to efficiently compute inner products of exponentially large density operators that are classically hard to compute. However, having an exponentially large feature space renders the problem of generalization hard. Furthermore, being able to evaluate inner products in high dimensional spaces efficiently by itself does not guarantee a quantum advantage, as already classically tractable kernels can correspond to high- or infinite-dimensional reproducing kernel Hilbert spaces (RKHS). We analyze the spectral properties of quantum kernels and find that we can expect an advantage if their RKHS is low dimensional and contains functions that are hard to compute classically. If the target function is known to lie in this class, this implies a quantum advantage, as the quantum computer can encode this inductive bias, whereas there is no classically efficient way to constrain the function class in the same way. However, we show that finding suitable quantum kernels is not easy because the kernel evaluation might require exponentially many measurements. In conclusion, our message is a somewhat sobering one: we conjecture that quantum machine learning models can offer speed-ups only if we manage to encode knowledge about the problem at hand into quantum circuits, while encoding the same bias into a classical model would be hard. These situations may plausibly occur when learning on data generated by a quantum process, however, they appear to be harder to come by for classical datasets.

翻译：假设量子计算机可以很好地用于机器学习。 在目前的工作中, 我们分析通过量子内核定义的函数类。 量子计算机提供了有效计算指数性大密度操作员的内产产品的可能性, 典型地难以计算。 但是, 具有指数性巨大的特性空间, 使一般化问题变得非常困难。 此外, 能够有效地评估高维空间的内产产品本身并不能保证量子优势, 因为已经古老可移植的内核可以与高或无限的再生内核Hilbert空间( RKHS) 相对应。 我们分析量子内核的光谱特性, 发现如果它们的量子内核运行是低度的, 并且含有难以进行典型性计算的功能。 如果目标功能是在本类中已知的, 则意味着量子优势, 因为量子计算机可以将这种感性偏差编码成感性偏差, 而用同样的方式来限制功能类。 然而, 我们显示, 找到合适的量内核内核的光核特性特性的特性特性特性, 当我们学习一种感官的直径直径直达的模型时, 需要一种直径直径直判的测, 。