I believe in quantum machine learning, it is interesting to talk about RKHS(reproducing kernel Hilbert space) and Hilbert space where a quantum state lives in. How do we think of these two spaces? Are they effectively the same thing in the context of quantum ML?
I think when talking about quantum machine learning, the notion of the RKHS and the Hilbert space where quantum states live are related but are two distinct spaces.
The RKHS is the space of the kernel functions $k(x^k,x)$ i.e. a space of symmetric and positive semi-definite functions. There is nothing particularly related to quantum computing about this. I would recommend this great lecture to learn more about the RKHS.
Now we get to the Hilbert space where the quantum states live. We can represent a kernel function $k(x^k,x)$ (living in the RKHS) by density matrices as (details in this paper):
$$k(x^k,x) = tr[\rho(x^k)\rho(x)]$$
where $\rho(x)$ are density matrices aka quantum states (whose form depends on $x$) that live in their own Hilbert space. And using the formula above, you can use these states to compute some $k(x^k,x)$ which lives in the (separate) RKHS.
But even though the quantum states and kernel functions live in well-defined separate Hilbert spaces, they are related: $\rho(x)$ lives in the space of the quantum states, and $k(x^k,x)$ lives in the space of the (linear combinations) of the trace of the product of different $\rho(x)$.
So the next question is: how can we understand this relation better? I.e. what kinds of interesting kernels (that live in the RKHS) can you efficiently create with quantum states (that live in their own Hilbert space)? This is currently an active area of research :) Hope this helps!