Hi Maria @Maria_Schuld,

Here is my question: where does the nonlinearity come from in QNN? My understanding is that:

- Feature embedding layer introduces nonlinearity.
- Unentangled parametrical gates do not introduce any nonlinearity, is this right?
- Entangling is nonlinear. But can it involve strong nonlinearity? Let’s say, by simply adopting several StronglyEntanglingLayers, is it possible to mimic a strongly nonlinear boundary in classification?
- Measurements introduce nonlinearity. I see this in papers but actually I didn’t understand the logic.

I would really appreciate it if you could briefly explain, or guide me to some referefences. Thanks in advance!

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?

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Thanks Maria @Maria_Schuld ! That really helps. A following-up question is that does the nonlinearity in parameters make sense for ML? Since ML looks for a mapping from input features *x* to output *y*, using of course the parameterized blocks (neurons or gates), I feel only the nonlinearity in *x* contributes. Please correct me if I am wrong. Thank you.

Yes, the nonlinearity in the parameters will have a more severe effect on the training landscape than the class of models one can learn…

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