The Inductive Bias of Quantum Kernels

TL;DR

This paper investigates the potential advantages of quantum kernels in machine learning, emphasizing that quantum speed-ups depend on encoding problem-specific knowledge into quantum circuits, especially when target functions are hard to compute classically.

Contribution

It analyzes the spectral properties of quantum kernels, highlighting conditions for quantum advantage and discussing the challenges in finding suitable kernels due to measurement complexity.

Findings

Quantum kernels can offer advantages if their RKHS is low-dimensional and contains classically hard functions.

Encoding problem-specific knowledge into quantum circuits is crucial for quantum speed-ups.

Finding suitable quantum kernels is challenging due to exponential measurement requirements.

Abstract

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.