Covariant quantum kernels for data with group structure

The use of kernel functions is a common technique to extract important features from datasets. A quantum computer can be used to estimate kernel entries as transition amplitudes of unitary circuits. Quantum kernels exist that, subject to computational hardness assumptions, cannot be computed classically. The learning problems for these cases are constructed artificially and it is an important challenge to find quantum kernels that have the potential to be relevant for real-world data. Here we identify a suitable class of learning problems on data that have a group structure, which are amenable to kernel methods. We introduce a family of quantum kernels that can be applied to such data, generalizing from a kernel that is known to have a quantum-classical separation when solving a particular set of problems. We use 27 qubits of a superconducting processor to demonstrate our method with a learning problem that embodies the structure of many essential learning problems on groups. The kernel method in machine learning can be implemented on near-term quantum computers. A 27-qubit device has now been used to solve learning problems using kernels that have the potential to be practically useful.