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I'm trying to understand quantum neural networks from reading Alchieri et al.'s review paper. The following paragraph describes the differences between classical and quantum neural networks:

Also, it is worth pointing out a key difference between classical and quantum neural networks: the former, in fact, are usually highly non linear models, while in quantum mechanics operators that act on states are always linear. Implementing a non-linear activation function in a quantum neural network is a major problem, and several attempts have been made to overcome it, e.g. by using specific measurements. To this day, though, there are no proposals for the implementation of a non-linear quantum operator, and most quantum neural networks offload nonlinearities to classical computers, or make use of quantum kernels.

Specifically, I want to clarify their statement:

"most quantum neural networks offload nonlinearities to classical computers, or make use of quantum kernels."

How does a quantum kernel handle the nonlinearities? I know generally how quantum kernels work, but I'm confused by the "nonlinearity" aspect that they describe.

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The quoted statement is a bit vague taken out of context, but a few comments on what they might have meant:

  1. Any quantum operation (meaning any physical evolution that is compatible with quantum mechanics; or more formally speaking, any quantum channel) is linear with respect to the input density matrix. This is intrinsic to quantum mechanics. Similarly, the relation between input density matrix and output probability distribution is always linear. Most quantum machine learning protocol rely on classically post-processing output probability distribution. So if those are linear in the input, if you want any sort of nonlinearity (which you most likely do) you need to inject it into the way you classically post-process the measurement data. In this sense, the sentence "there are no proposals for the implementation of a non-linear quantum operator", at least out of context, is somewhat misleading IMO: it's not that there are no proposals for such implementations; it's that a quantum operation cannot be nonlinear in the input density matrix. Unless they mean something else with "non-linear quantum operation", of course.

  2. "or make use of quantum kernels" is also a very vague remark, but it might refer to the fact that my statements above do not take into account the fact that if you want to use QML to process classical data, then you don't really directly care about the linearity of the relation between input density matrix and output probability distribution, but rather about the linearity of the relation between the input classical information and the output probability distribution. Such relation passes through some (typically nonlinear) way of encoding the classical information into the quantum states. This is often referred to as the "encoding map" or "feature map", and is a chief subject of what people study in the context of quantum kernels.

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