Temel Kavramlar
All kernel functions can be approximated as embedding quantum kernels, and efficient embedding quantum kernels are universal within the classes of shift-invariant and composition kernels.
Özet
The key insights from the content are:
Universality of Embedding Quantum Kernels (EQKs):
Any kernel function can be approximated as an EQK, using a finite-dimensional quantum feature map.
This is an existence result, without claims about the practicality or efficiency of the construction.
Efficient Approximations of Shift-Invariant Kernels as EQKs:
Shift-invariant kernels can be efficiently approximated as EQKs, provided they are smooth enough.
The authors use random Fourier features to construct a space-efficient EQK approximation, and provide sufficient conditions for a time-efficient construction.
Efficient Approximations of Composition Kernels as EQKs:
The authors introduce a new class of "composition kernels" that generalize shift-invariant kernels.
They prove that efficient EQKs are universal within the class of efficient composition kernels, which includes the "projected quantum kernel" from prior work.
Overall, the content establishes the expressivity of EQKs, and identifies two important classes of kernels that can be efficiently approximated using EQKs.