Hey @zhouyf10, and thanks for migrating the question to a new thread!
I assume you mean nonlinearity of the QML model (say, the expectation of a variational circuit + measurement) in the data, right?
Let’s be precise and go through each step. If you embed inputs, creating a quantum state |\phi(x) \rangle, then the amplitudes \phi_i = \langle i | \phi(x)\rangle are in general nonlinear in x, yes. The obvious exception would be AmplitudeEmbedding
, which is designed so that the amplitudes are the inputs.
If you then do some more unitary evolution, for example by applying a parametrized circuit U(\theta), the amplitudes of the new state |\phi'(x) \rangle = U(\theta)| \phi(x) \rangle will be linear in the amplitudes of |\phi(x) \rangle (that’s trivial, since U is a linear operation on the state vector). So any standard computation after the embedding does not add nonlinearity, which means that embedded inputs have the same distance to each other before and after U is applied. Whether there is entanglement or not does not play a role at all.
Finally, the measurement will again introduce a “slight” (quadratic) nonlinearity in the amplitudes of the final state: \langle \phi'(x)| M|\phi'(x) \rangle = \sum_{ij} M_{ij} \phi'_i(x) \phi'_j(x).
In this sense one can say that the source of nonlinearity with regards to the inputs is only the embedding and the measurement. But obviously a parametrized gate like a Pauli rotation produces a state which is nonlinear in the parameters. So a sentence like “xyz is (non)linear” only makes sense if we state what object is (non)linear in what variable…
Hope this helps?