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Will non-linear quantum computers ever be possible? If so, how would they be made, and what would programming them be like? Do we still model non-linear states with vectors and transformations with unitary matrices?

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If you mean, quantum computers that would exploit a nonlinear term in the Schrödinger equation to solve NP-complete and even harder problems in polynomial time (as studied by Abrams and Lloyd in 1998), then the central problem is that I don’t believe such a nonlinear term exists. Experimental searches have put more and more severe upper bounds on its size, if it did exist. But the bigger problem is that such a term would completely break the structure of quantum mechanics, in even more “basic” ways than by making NP-complete problems easy. For example, it would genetically lead to faster-than-light communication and to violations of the uncertainty principle. So while it’s interesting to think about, and even worth searching for just in case, it almost certainly belongs to the category of “what if?” physics.

What does "linear" mean to you? Integration is linear[1]. The Hilbert space of all the functions you care about is linear[2]. Life is locally linear[3]. So maybe "non-linear" is just a red herring.

1. https://en.wikipedia.org/wiki/Linearity_of_integration

2. Not really but close enough.

3. I don't know what I mean either, but https://en.wikipedia.org/wiki/Linear_time-invariant_theory is interesting anyway.

I mean quantum computers that harness the NLSE in their computations somehow.

Interesting. That's way more physics than I know. :)

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